‫ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‪ ١‬ﻭ ﻣﺤﺎﺳﺒﻪ ﺁﻥ ﺩﺭ ﺑﺎﺯﯼﻫﺎﯼ ﭼﻨﺪ ﺑﺎﺯﻳﮑﻨﯽ‬
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‫ﺳﻴﺪﺍﺑﻮﺍﻟﻘﺎﺳﻢ ﻣﻴﺮﺭﻭﺷﻨﺪﻝ‬
‫‪Mirroshandel@ce.sharif.edu‬‬
‫ﭼﮑﻴﺪﻩ‪ .‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‪ ،‬ﻳﮏ ﻣﻔﻬﻮﻡ ﺍﺳﺘﺎﻧﺪﺍﺭﺩ ﺑﺮﺍﯼ ﻧﺸﺎﻥ ﺩﺍﺩﻥ ﻋﻘﻼﻳﻲ ﺑﻮﺩﻥ ﻳﮏ ﺑﺎﺯﯼ ﺍﺳﺖ ﮐﻪ ﻳﮏ‬
‫ﺗﻮﺯﻳﻊ ﻋﻤﻮﻣﯽ ﺭﺍ ﺩﺭ ﻧﻤﺎﻳﻪﻫﺎﯼ ﺍﺳﺘﺮﺍﺗﮋﯼ‪ ٣‬ﻧﺸﺎﻥ ﻣﯽﺩﻫﺪ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ‪ ،‬ﺍﺑﺘﺪﺍ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﻣﻌﺮﻓﯽ‬
‫ﻣﯽﺷﻮﺩ ﻭ ﺳﭙﺲ ﻳﮏ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺑﺎ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ‪ ٤‬ﺍﺭﺍﺋﻪ ﺧﻮﺍﻫﺪ ﺷﺪ ﮐﻪ ﻣﯽﺗﻮﺍﻥ ﺁﻥ ﺭﺍ ﺩﺭ‬
‫ﺑﺎﺯﻱﻫﺎﻳﻲ ﮐﻪ ﺑﻪ ﺻﻮﺭﺕ ﻓﺸﺮﺩﻩ‪ ٥‬ﻗﺎﺑﻞ ﺍﺭﺍﺋﻪ ﻫﺴﺘﻨﺪ‪ ،‬ﺍﻋﻤﺎﻝ ﻭ ﺣﺪﺍﻗﻞ ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﺎﺯﯼ ﺭﺍ ﺑﻪ‬
‫ﺻﻮﺭﺕ ﮐﺎﺭﺍ ﭘﻴﺪﺍ ﻧﻤﻮﺩ‪ .‬ﺍﺯ ﺍﻳﻦ ﺩﺳﺘﻪ ﺍﺯ ﺑﺎﺯﯼﻫﺎ ﻣﯽﺗﻮﺍﻥ ﺑﻪ ﺑﺎﺯﯼﻫﺎﻳﻲ ﻧﻈﻴﺮ ﺑﺎﺯﯼﻫﺎﯼ ﮔﺮﺍﻓﻴﮑﯽ‪ ،‬ﺑﺎﺯﯼﻫﺎﯼ ﭼﻨﺪ‬
‫ﻣﺎﺗﺮﻳﺴﯽ‪ ،٦‬ﺑﺎﺯﯼﻫﺎﯼ ﺑﯽﻧﺎﻡ‪ ٧‬ﻭ ﺑﺎﺯﯼﻫﺎﯼ ﺯﻣﺎﻥﺑﻨﺪﯼ ﺍﺷﺎﺭﻩ ﻧﻤﻮﺩ‪ .‬ﺩﺭ ﺍﺩﺍﻣﻪ ﺑﺮﺍﯼ ﺍﻳﻦ ﺑﺎﺯﯼﻫﺎ‪ ،‬ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ‬
‫ﻣﺤﺎﺳﺒﻪ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ ﻣﻮﺭﺩ ﺑﺮﺭﺳﯽ ﻗﺮﺍﺭ ﺧﻮﺍﻫﺪ ﮔﺮﻓﺖ ﻭ ﺧﻮﺍﻫﻴﻢ ﺩﻳﺪ ﮐﻪ ﺩﺭ ﻣﺠﻤﻮﻉ ﭘﻴﺪﺍ ﻧﻤﻮﺩﻥ‬
‫ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ ﺩﺍﺭﺍﯼ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ ‪ NP-hard‬ﺍﺳﺖ‪.‬‬
‫ﮐﻠﻤﺎﺕ ﮐﻠﻴﺪﯼ‪ :‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ‪ ،‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‪ ،‬ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ‪ ،‬ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﻭ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‬
‫ﺑﻬﻴﻨﻪ‬
‫‪ ١‬ﻣﻘﺪﻣﻪ‬
‫ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ‪ ،‬ﻳﮏ ﻣﻔﻬﻮﻡ ﺍﺳﺘﺎﻧﺪﺍﺭﺩ ﺩﺭ ﻣﻮﺭﺩ ﻋﻘﻼﻳﻲ ﺑﻮﺩﻥ ﺩﺭ ﻧﻈﺮﻳﻪ ﺑﺎﺯﯼﻫﺎ ﺍﺳﺖ‪ .‬ﻣﯽﺗﻮﺍﻥ ﺳﻪ ﺩﻟﻴﻞ ﺑﺮﺍﯼ ﺍﻳﻦ‬
‫ﺍﺳﺘﺎﻧﺪﺍﺭﺩ ﺑﻮﺩﻥ ﺍﺭﺍﺋﻪ ﻧﻤﻮﺩ‪ :‬ﻫﻤﻪ ﺍﺯ ﺍﻳﻦ ﻣﻔﻬﻮﻡ ﺍﺳﺘﻔﺎﺩﻩ ﻣﯽﮐﻨﻨﺪ؛ ﺑﺴﻴﺎﺭﯼ ﺍﺯ ﺑﻬﺒﻮﺩﻫﺎ ﻭ ﺗﻌﻤﻴﻢﻫﺎ ﺑﺎ ﺍﻳﻦ ﻣﻔﻬﻮﻡ ﻣﻘﺎﻳﺴﻪ‬
‫ﻣﯽﺷﻮﻧﺪ؛ ﻭ ﺩﺭ ﻧﻬﺎﻳﺖ‪ ،‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ ﺑﺎ ﺩﺍﺷﺘﻦ ﺑﻌﻀﯽ ﺍﺯ ﻣﺴﺎﺋﻞ ﻣﺮﺗﺒﻂ ﮐﻪ ﻫﻤﭽﻨﺎﻥ ﺟﺰ ﻣﺴﺎﺋﻞ ﺑﺎﺯ‪ ٨‬ﻣﺤﺴﻮﺏ‬
‫ﻣﯽﺷﻮﻧﺪ‪ ،‬ﺑﻪ ﻋﻨﻮﺍﻥ ﻳﮏ ﻭﺍﺳﻂ ﺑﻴﻦ ﻧﻈﺮﻳﻪ ﺑﺎﺯﯼﻫﺎ ﻭ ﺍﻟﮕﻮﺭﻳﺘﻢﻫﺎ ﻋﻤﻞ ﻣﯽﮐﻨﺪ )ﻫﻤﺎﻧﻄﻮﺭ ﮐﻪ ﺍﺧﻴﺮﺍ ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‬
‫ﻣﺤﺎﺳﺒﻪ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ ﺗﺮﮐﻴﺒﯽ‪ ٩‬ﺣﺘﯽ ﺩﺭ ﻳﮏ ﺑﺎﺯﯼ ﺩﻭ ﻧﻔﺮﻩ ‪ PPAD-Complete‬ﺍﺳﺖ ]‪.([۱‬‬
‫ﻫﻤﺎﻧﻄﻮﺭ ﮐﻪ ﮔﻔﺘﻪ ﺷﺪ‪ ،‬ﭼﻨﺪﻳﻦ ﺑﻬﺒﻮﺩ ﻭ ﺗﻌﻤﻴﻢ ﺑﺮﺍﯼ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ ﺑﻮﺟﻮﺩ ﺁﻣﺪﻩ ﺍﺳﺖ‪ .‬ﺍﺻﻠﯽﺗﺮﻳﻦ ﺁﻧﻬﺎ‪ ،‬ﻧﻘﻄﻪ‬
‫ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺍﺳﺖ‪ .‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ ﺗﺮﮐﻴﺒﯽ‪ ،‬ﻳﮏ ﺗﻮﺯﻳﻊ ﺩﺭ ﻓﻀﺎﯼ ﺍﺳﺘﺮﺍﺗﮋﯼﻫﺎﺳﺖ ﮐﻪ ﻧﺎﻫﻤﺒﺴﺘﻪ ﺍﺳﺖ )ﺯﻳﺮﺍ‬
‫ﺣﺎﺻﻠﻀﺮﺏ ﺍﺣﺘﻤﺎﻝ ﺗﻮﺯﻳﻊﻫﺎﯼ ﻣﺴﺘﻘﻞ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺳﺖ(‪ ،‬ﺩﺭ ﺣﺎﻟﯽ ﮐﻪ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‪ ،‬ﻳﮏ ﺗﻮﺯﻳﻊ ﻋﻤﻮﻣﯽ ﺩﺭ‬
‫ﻧﻤﺎﻳﻪﻫﺎﯼ ﺍﺳﺘﺮﺍﺗﮋﯼ ﺍﺳﺖ‪ .‬ﺍﻟﺒﺘﻪ ﮐﺎﻣﻼ ﻣﺸﺨﺺ ﺍﺳﺖ ﮐﻪ ﺍﻳﻦ ﻧﻘﻄﻪ‪ ،‬ﺩﺍﺭﺍﯼ ﺧﺼﻮﺻﻴﺖ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﺍﺳﺖ‪ :‬ﺍﮔﺮ ﻳﮏ ﻧﻤﺎﻳﻪ‬
‫ﺍﺳﺘﺮﺍﺗﮋﯼ ﺍﺯ ﺍﻳﻦ ﺗﻮﺯﻳﻊ ﺍﻧﺘﺨﺎﺏ ﺷﻮﺩ )ﺍﺣﺘﻤﺎﻻ ﺍﻳﻦ ﮐﺎﺭ ﺗﻮﺳﻂ ﻳﮏ ﻋﺎﻣﻞ ﺧﺎﺭﺟﯽ ﻣﻮﺭﺩ ﺍﻋﺘﻤﺎﺩ ﻃﺮﻓﻴﻦ ﺍﻧﺠﺎﻡ ﺧﻮﺍﻫﺪ‬
‫ﺷﺪ ﻭ ﺍﻳﻦ ﺍﺳﺘﺮﺍﺗﮋﯼ ﺑﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺑﻪ ﺻﻮﺭﺕ ﺟﺪﺍﮔﺎﻧﻪ ﺍﻋﻼﻡ ﻣﯽﺷﻮﺩ(‪ ،‬ﺁﻧﮕﺎﻩ ﻫﻴﭻ ﻳﮏ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺗﻤﺎﻳﻠﯽ ﺑﻪ ﺍﻧﺘﺨﺎﺏ‬
‫ﺍﺳﺘﺮﺍﺗﮋﯼ ﺩﻳﮕﺮ ﻧﺪﺍﺭﻧﺪ‪ ،‬ﺯﻳﺮﺍ ﺑﺎ ﺍﻳﻦ ﻓﺮﺽ ﮐﻪ ﺑﻘﻴﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺯ ﺍﻳﻦ ﺍﺳﺘﺮﺍﺗﮋﯼ ﺗﺒﻌﻴﺖ ﮐﻨﻨﺪ‪ ،‬ﺍﺳﺘﺮﺍﺗﮋﯼ ﭘﻴﺸﻨﻬﺎﺩﯼ‪،‬‬
‫ﻣﻨﺎﺳﺐﺗﺮﻳﻦ ﻣﻨﻔﻌﺖ ﺭﺍ ﺑﻪ ﻫﻤﺮﺍﻩ ﺧﻮﺍﻫﺪ ﺩﺍﺷﺖ‪.‬‬
‫ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺩﺍﺭﺍﯼ ﭼﻨﺪﻳﻦ ﺧﺼﻮﺻﻴﺖ ﻣﻬﻢ ﻭ ﺩﺭ ﻋﻴﻦ ﺣﺎﻝ ﻣﻔﻴﺪ ﺍﺳﺖ‪ :‬ﺍﻳﻦ ﻧﻘﻄﻪ‪ ،‬ﻳﮏ ﻣﻔﻬﻮﻡ ﮐﺎﻣﻼ‬
‫ﻣﻨﻄﻘﯽ‪ ،‬ﺳﺎﺩﻩ ﻭ ﻣﺤﺘﻤﻞ ﺍﺳﺖ؛ ﺩﺭ ﻣﻮﺭﺩ ﺑﺎﺯﯼﻫﺎﻳﻲ ﺑﺎ ﺗﻌﺪﺍﺩ ﺑﺎﺯﻳﮑﻦ ﻣﺤﺪﻭﺩ ﻭ ﺗﻌﺪﺍﺩ ﮐﻨﺶﻫﺎﯼ ﻣﺤﺪﻭﺩ ﺣﺘﻤﺎ ﻣﻮﺟﻮﺩ‬
‫ﺍﺳﺖ )ﺯﻳﺮﺍ ﺑﻪ ﺳﺎﺩﮔﯽ ﻣﯽﺗﻮﺍﻥ ﻓﻬﻤﻴﺪ ﮐﻪ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺍﺳﺖ(؛ ﺩﺭ ﻋﻤﻞ ﺩﺍﺭﺍﯼ ﺳﺎﺩﮔﯽ ﻭ‬
‫ﻃﺒﻴﻌﯽ ﺑﻮﺩﻧﯽ ﺍﺳﺖ ﮐﻪ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ ﻓﺎﻗﺪ ﺁﻥ ﺍﺳﺖ ]‪[۲‬؛ ﻭ ﻣﯽﺗﻮﺍﻥ ﺁﻥ ﺭﺍ ﺩﺭ ﺯﻣﺎﻥ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ‬
‫ﺑﺮﻧﺎﻣﻪﺳﺎﺯﯼ ﺧﻄﯽ‪ ١٠‬ﺑﺪﺳﺖ ﺁﻭﺭﺩ‪ ،‬ﺯﻳﺮﺍ ﻧﺎﻣﺴﺎﻭﯼﻫﺎﻳﻲ ﮐﻪ ﺑﺮﺍﯼ ﻧﺸﺎﻥ ﺩﺍﺩﻥ ﺧﺼﻮﺻﻴﺖ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﮑﺎﺭ‬
‫ﻣﯽﺭﻭﻧﺪ‪ ،‬ﺧﻄﯽ ﻫﺴﺘﻨﺪ‪.‬‬
‫ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﺻﻠﯽ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ ﺑﺮ ﺍﺳﺎﺱ ﺗﻐﻴﻴﺮ ﺩﺭ ﺍﺛﺒﺎﺕ ﻭﺟﻮﺩ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺩﺭ ]‪ [٣‬ﻋﻤﻞ ﻣﯽﮐﻨﺪ‪ .‬ﺩﺭ ﺍﻳﻦ‬
‫ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﺯ ﺗﺮﮐﻴﺐ ﺩﻭﮔﺎﻧﮕﯽ ﺑﺮﻧﺎﻣﻪﺳﺎﺯﯼ ﺧﻄﯽ‪ ،١١‬ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﻟﻴﭙﺴﻮﻳﺪ‪ ١٢‬ﻭ ﻣﺤﺎﺳﺒﺎﺕ ﻣﺮﺑﻮﻁ ﺑﻪ ﻭﺿﻌﻴﺖ ﭘﺎﻳﺪﺍﺭ‪ ١٣‬ﺩﺭ‬
‫ﺯﻧﺠﻴﺮﻩ ﻣﺎﺭﮐﻮﻑ ﺍﺳﺘﻔﺎﺩﻩ ﻣﯽﺷﻮﺩ ﺗﺎ ﺑﺘﻮﺍﻥ ﺑﻌﻀﯽ ﺍﺯ ﻧﻘﺎﻁ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺭﺍ ﭘﻴﺪﺍ ﻧﻤﻮﺩ‪.‬‬
‫ﺣﺎﻝ ﻳﮏ ﺳﻮﺍﻝ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﻣﺤﺎﺳﺒﻪ ﭘﻴﭽﻴﺪﮔﯽ ﻧﻘﺎﻁ ﺗﻌﺎﺩﻝ ﭼﻪ ﺍﻫﻤﻴﺘﯽ ﺩﺍﺭﺩ؟ ﭘﺎﺳﺦ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﻧﻘﺎﻁ ﺗﻌﺎﺩﻝ‬
‫ﻧﺸﺎﻥ ﺩﻫﻨﺪﻩ ﻋﻘﻼﻳﻲ ﺑﻮﺩﻥ ﺑﺎﺯﯼ ﻫﺴﺘﻨﺪ‪ ،‬ﻣﺪﻝﻫﺎﻳﻲ ﺭﺍ ﺑﺮﺍﯼ ﺭﻓﺘﺎﺭ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺭﺍﺋﻪ ﻣﯽﮐﻨﻨﺪ ﻭ ﻧﺸﺎﻥﺩﻫﻨﺪﻩ ﻗﺪﺭﺕ ﻣﺪﻝ‬
‫ﻧﻤﻮﺩﻥ ﺑﺎﺯﯼ ﻫﺴﺘﻨﺪ ]‪ .[۲ ،۱‬ﺩﺭ ﻭﺍﻗﻊ ﻗﺪﺭﺕ ﻣﺤﺎﺳﺒﻪ ﺩﺭ ﻣﻮﺭﺩ ﻳﮏ ﻣﺴﺎﻟﻪ‪ ،‬ﻣﻴﺰﺍﻥ ﺳﻮﺩﻣﻨﺪﯼ ﺁﻥ ﻣﺴﺎﻟﻪ ﺭﺍ ﺩﺭ ﮐﺎﺭﺑﺮﺩﻫﺎ‬
‫ﻧﺸﺎﻥ ﻣﯽﺩﻫﺪ‪ ،‬ﻫﻤﺎﻧﻄﻮﺭ ﮐﻪ ﻳﮑﯽ ﺍﺯ ﺑﺰﺭﮔﺎﻥ ﻋﻠﻮﻡ ﮐﺎﻣﭙﻴﻮﺗﺮ ﮔﻔﺘﻪ ﺍﺳﺖ‪" :‬ﺍﮔﺮ ﮐﺎﻣﭙﻴﻮﺗﺮ ﺷﻤﺎ ﻧﺘﻮﺍﻧﺪ ﺣﻞ ﻳﮏ ﻣﺴﺎﻟﻪ ﺭﺍ ﭘﻴﺪﺍ‬
‫ﮐﻨﺪ‪ ،‬ﺑﺎﺯﺍﺭ ﻗﻄﻌﺎ ﻧﺨﻮﺍﻫﺪ ﺗﻮﺍﻧﺴﺖ" ]‪.[۱‬‬
‫ﺩﺭ ﺍﺩﺍﻣﻪ ﺍﻳﻦ ﻣﻘﺎﻟﻪ‪ ،‬ﺩﺭ ﺑﺨﺶ ‪ ،۲‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺩﻗﻴﻖﺗﺮ ﺑﺮﺭﺳﯽ ﺷﺪﻩ ﻭ ﺗﻌﺮﻳﻒ ﺭﺳﻤﯽ ﺁﻥ ﺍﺭﺍﺋﻪ ﺧﻮﺍﻫﺪ ﺷﺪ‪.‬‬
‫ﺳﭙﺲ ﺩﺭ ﺑﺨﺶ ‪ ،۳‬ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﺑﻪ ﺻﻮﺭﺕ ﺍﺟﻤﺎﻟﯽ ﻣﻌﺮﻓﯽ ﺷﺪﻩ ﻭ ﭼﻨﺪ ﻣﺜﺎﻝ ﺍﺯ ﺁﻧﻬﺎ ﺷﺮﺡ ﺩﺍﺩﻩ ﺧﻮﺍﻫﺪ ﺷﺪ‪ .‬ﺩﺭ‬
‫ﺑﺨﺶ ‪ ،۴‬ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﺑﺘﺪﺍﻳﻲ ﭘﻴﺪﺍ ﻧﻤﻮﺩﻥ ﺑﻌﻀﯽ ﺍﺯ ﻧﻘﺎﻁ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺍﺭﺍﺋﻪ ﻣﯽﺷﻮﺩ ﻭ ﺩﺭ ﺑﺨﺶ ‪ ،۵‬ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﻣﯽﺷﻮﺩ‬
‫ﮐﻪ ﭘﻴﺪﺍ ﮐﺮﺩﻥ ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﺮﺍﯼ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﺩﺍﺭﺍﯼ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺍﺳﺖ‪ .‬ﺍﻟﮕﻮﺭﻳﺘﻢ‬
‫ﻣﺮﺗﺒﻂ ﺑﺮﺍﯼ ﭘﻴﺪﺍ ﮐﺮﺩﻥ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ ﻭ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ ﻣﺮﺗﺒﻂ ﺑﺎ ﺁﻥ ﺩﺭ ﺑﺨﺶ ‪ ۶‬ﺍﺭﺍﺋﻪ ﺧﻮﺍﻫﺪ ﺷﺪ‪ .‬ﺩﺭ ﺍﻧﺘﻬﺎ‬
‫ﺩﺭ ﺑﺨﺶ ‪ ،۷‬ﺑﻪ ﻧﺘﻴﺠﻪ ﮔﻴﺮﻱ ﻣﻲﭘﺮﺩﺍﺯﻳﻢ‪.‬‬
‫‪ ٢‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‬
‫ﺩﺭ ﻧﻈﺮﻳﻪ ﺑﺎﺯﯼﻫﺎ‪ ،‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﻳﮏ ﻣﻔﻬﻮﻡ ﺣﻞ ﻣﺴﺎﻟﻪ ﺍﺳﺖ ﮐﻪ ﺍﺯ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻣﺸﻬﻮﺭ ﻧﺶ‪ ،‬ﺑﺴﻴﺎﺭ ﻋﻤﻮﻣﯽﺗﺮ‬
‫ﺍﺳﺖ‪ .‬ﺍﻳﻦ ﻣﻔﻬﻮﻡ ﺍﻭﻟﻴﻦ ﺑﺎﺭ ﺩﺭ ﺳﺎﻝ ‪ ۱۹۷۴‬ﺗﻮﺳﻂ ﺭﻳﺎﺿﻴﺪﺍﻧﯽ ﺑﻪ ﻧﺎﻡ ﺭﺍﺑﺮﺕ ﺍﻭﻣﺎﻥ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ ]‪ .[۴‬ﺍﻳﺪﻩ ﺍﺻﻠﯽ ﺍﻳﻦ‬
‫ﺍﺳﺖ ﮐﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ‪ ،‬ﮐﻨﺶ ﻣﻮﺭﺩ ﻧﻈﺮ ﺧﻮﺩ ﺭﺍ ﺑﺮ ﺍﺳﺎﺱ ﺩﺭﻳﺎﻓﺘﺶ ﺍﺯ ﻳﮏ ﻋﻼﻣﺖ‪ ١٤‬ﻋﻤﻮﻣﯽ ﺍﻧﺠﺎﻡ ﻣﯽﺩﻫﺪ‪ .‬ﺩﺭ‬
‫ﺍﺳﺘﺮﺍﺗﮋﯼ‪ ،‬ﺑﺎﺯﺍﯼ ﻫﺮ ﺩﺭﻳﺎﻓﺘﯽ ﮐﻪ ﺍﻧﺠﺎﻡ ﻣﯽﺷﻮﺩ‪ ،‬ﻳﮑﯽ ﺍﺯ ﮐﻨﺶﻫﺎﯼ ﻣﻤﮑﻦ ﺑﺎﺯﻳﮑﻦ ﺗﺨﺼﻴﺺ ﺩﺍﺩﻩ ﻣﯽﺷﻮﺩ‪ .‬ﺣﺎﻝ ﺍﮔﺮ‬
‫ﻫﻴﭻ ﺑﺎﺯﻳﮑﻨﯽ ﻧﺨﻮﺍﻫﺪ ﺍﺯ ﺍﺳﺘﺮﺍﺗﮋﯼ ﭘﻴﺸﻨﻬﺎﺩﯼ ﺳﺮﭘﻴﭽﯽ ﮐﻨﺪ‪ ،‬ﺑﻪ ﺍﻳﻦ ﺗﻮﺯﻳﻊ ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﮔﻔﺘﻪ ﻣﯽﺷﻮﺩ‪ .‬ﺩﺭ‬
‫ﺍﺩﺍﻣﻪ ﺗﻌﺮﻳﻒ ﺭﺳﻤﯽ ﺍﻳﻦ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻣﻄﺮﺡ ﺧﻮﺍﻫﺪ ﺷﺪ ﻭ ﺳﭙﺲ ﻣﺜﺎﻝﻫﺎﻳﻲ ﺍﺯ ﺁﻥ ﺍﺭﺍﺋﻪ ﻣﯽﺷﻮﺩ‪.‬‬
‫‪ ١,٢‬ﺗﻌﺮﻳﻒ ﺭﺳﻤﯽ‬
‫ﻳﮏ ﺑﺎﺯﯼ ﮐﻪ ﺗﻌﺪﺍﺩ ﻣﺘﻨﺎﻫﯽ ﺑﺎﺯﻳﮑﻦ ﺩﺍﺭﺩ )‪ n‬ﺑﺎﺯﻳﮑﻦ‪ (1, 2, …, n :‬ﺭﺍ ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﺪ‪ .‬ﻓﺮﺽ ﮐﻨﻴﺪ ﮐﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ‬
‫‪ ، p ≤ n‬ﺩﺍﺭﺍﯼ ﺗﻌﺪﺍﺩ ﻣﺘﻨﺎﻫﯽ ﺍﺯ ﺍﺳﺘﺮﺍﺗﮋﯼﻫﺎ ﻳﺎ ﺍﻧﺘﺨﺎﺏﻫﺎ ﺑﺎﺷﺪ ) ‪ S p‬ﺍﺳﺘﺮﺍﺗﮋﯼ ﺑﺎﺯﻳﮑﻦ ‪ p‬ﺍﻡ ﺭﺍ ﻧﺸﺎﻥ ﻣﯽﺩﻫﺪ ﻭ ﺑﺮﺍﯼ‬
‫ﻫﺮ ﺑﺎﺯﻳﮑﻦ ‪ p‬ﺩﺍﺭﻳﻢ‪ .( S p ≥ 2 :‬ﺑﻪ ﻣﺠﻤﻮﻋﻪ ‪ ، S = ∏n S p‬ﻧﻤﺎﻳﻪﻫﺎﯼ ﺍﺳﺘﺮﺍﺗﮋﯼ ﺍﻃﻼﻕ ﻣﯽﺷﻮﺩ‪ .‬ﻣﺠﻤﻮﻋﻪ‬
‫‪p =1‬‬
‫‪Sq‬‬
‫‪n‬‬
‫‪q≠ p‬‬
‫∏‬
‫ﺑﺎ ‪ S − p‬ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﻣﯽﺷﻮﺩ‪ m .‬ﻫﻢ ﻧﺸﺎﻥ ﺩﻫﻨﺪﻩ ﺑﻴﺸﻴﻨﻪ ‪ S p‬ﺑﺮﺍﯼ ﺗﻤﺎﻣﯽ ﺑﺎﺯﻳﮑﻨﺎﻥ ‪ p‬ﺍﺳﺖ‪ .‬ﺗﺎﺑﻊ ﻣﻨﻔﻌﺖ‬
‫‪١٥‬‬
‫‪p‬‬
‫ﺑﺎﺯﻳﮑﻦ ‪ p‬ﻫﻢ ﺑﺎ ‪ u‬ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﻣﯽﺷﻮﺩ ﮐﻪ ﻳﮏ ﻧﮕﺎﺷﺖ ﺍﺯ ‪ S‬ﺑﻪ ﺍﻋﺪﺍﺩ ﻃﺒﻴﻌﯽ ﺍﺳﺖ‪.‬‬
‫ﻳﮏ ﺗﻮﺯﻳﻊ ﺭﻭﯼ ‪ ،S‬ﻳﮏ ﺑﺮﺩﺍﺭ ﻧﺎﻣﻨﻔﯽ ﺍﺯ ﺍﻋﺪﺍﺩ ﺣﻘﻴﻘﯽ ﺍﺳﺖ ﮐﻪ ﻫﺮ ﮐﺪﺍﻡ ﺍﺯ ﺍﻳﻦ ﺍﻋﺪﺍﺩ ﺑﺮﺍﯼ ﻳﮏ ﻧﻤﺎﻳﻪ ﺍﺳﺘﺮﺍﺗﮋﯼ ﺩﺭ‬
‫‪ S‬ﺑﮑﺎﺭ ﻣﯽﺭﻭﺩ ﻭ ﺣﺎﺻﻞ ﺟﻤﻊ ﺍﻳﻦ ﺍﻋﺪﺍﺩ ﺑﺮﺍﺑﺮ ‪ ۱‬ﺍﺳﺖ‪ .‬ﻳﮏ ﺗﻮﺯﻳﻊ ‪ x‬ﺭﻭﯼ ‪ S‬ﺩﺍﺭﺍﯼ ﺧﺎﺻﻴﺖ ﺣﺎﺻﻠﻀﺮﺑﯽ ﺍﺳﺖ ﺍﮔﺮ‬
‫ﺑﺮﺍﯼ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ‪ ،p‬ﻳﮏ ﺗﻮﺯﻳﻊ ‪ x p‬ﺩﺭ ‪ S p‬ﺑﺮﺍﯼ ﺗﻤﺎﻣﯽ ) ‪ s = (s1 , s 2 , ..., s n‬ﻭﺟﻮﺩ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ ﻭ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪:‬‬
‫‪p‬‬
‫‪Sp‬‬
‫‪x‬‬
‫‪n‬‬
‫‪p =1‬‬
‫∏‬
‫= ‪xs‬‬
‫ﺣﺎﻝ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻪ ﺍﻳﻦ ﺻﻮﺭﺕ ﺗﻌﺮﻳﻒ ﻣﯽﺷﻮﺩ‪ :‬ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﻳﮏ ﺗﻮﺯﻳﻊ ‪ x‬ﺭﻭﯼ ‪ S‬ﺍﺳﺖ‪ ،‬ﺑﻪ‬
‫ﻧﺤﻮﯼ ﮐﻪ ﺑﺮﺍﯼ ﺗﻤﺎﻣﯽ ﺑﺎﺯﻳﮑﻨﺎﻥ ‪ p‬ﻭ ﺗﻤﺎﻣﯽ ﺍﺳﺘﺮﺍﺗﮋﯼﻫﺎﯼ ‪ i, j ∈ S p‬ﺩﺍﺷﺘﻪ ﺑﺎﺷﻴﻢ‪ :‬ﺍﮔﺮ ﺍﺳﺘﺮﺍﺗﮋﯼ ﺍﻧﺘﺨﺎﺏ ﺷﺪﻩ ﺍﺯ ‪x‬‬
‫ﺑﺮﺍﯼ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ‪ ،p‬ﺑﺮﺍﺑﺮ ‪ i‬ﺑﺎﺷﺪ‪ ،‬ﻣﻨﻔﻌﺖ ﻣﻮﺭﺩ ﺍﻧﺘﻈﺎﺭ ‪ p‬ﺑﺎ ﺑﺎﺯﯼ ﮐﺮﺩﻥ ‪ i‬ﮐﻤﺘﺮ ﺍﺯ ﺑﺎﺯﯼ ﮐﺮﺩﻥ ‪ j‬ﻧﺸﻮﺩ‪:‬‬
‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻣﯽﺗﻮﺍﻥ ﺑﻪ ﺻﻮﺭﺕ ﺷﻬﻮﺩﯼ ﺑﺮ ﺍﺳﺎﺱ ﺗﻌﺮﻳﻒ ﺑﺎﻻ ﮔﻔﺖ ﮐﻪ ﺩﺭ ﺻﻮﺭﺗﯽ ﮐﻪ ﻳﮏ ﻋﺎﻣﻞ ﻣﻮﺭﺩ ﺍﻃﻤﻴﻨﺎﻥ‪ ،‬ﻳﮏ‬
‫ﻧﻤﺎﻳﻪ ﺍﺳﺘﺮﺍﺗﮋﯼ ‪ s‬ﺭﺍ ﺍﺯ ﺗﻮﺯﻳﻊ ﺍﻧﺘﺨﺎﺏ ﮐﻨﺪ ﻭ ﺑﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺑﻪ ﺻﻮﺭﺕ ﻣﺠﺰﺍ ﺍﻋﻼﻡ ﻧﻤﺎﻳﺪ‪ ،‬ﺩﻳﮕﺮ ﻫﻴﭻ ﺑﺎﺯﻳﮑﻨﯽ ﺗﻤﺎﻳﻠﯽ ﺑﻪ‬
‫ﺗﻐﻴﻴﺮ ﺍﺳﺘﺮﺍﺗﮋﯼ ﺧﻮﺩ ﻧﺨﻮﺍﻫﺪ ﺩﺍﺷﺖ )ﻳﻌﻨﯽ ﺩﺭ ﻣﻮﺭﺩ ﺑﺎﻻ‪ ،‬ﺑﺎﺯﻳﮑﻦ ‪ ،p‬ﺍﺳﺘﺮﺍﺗﮋﯼ ‪ j‬ﺭﺍ ﺑﻪ ﺍﺳﺘﺮﺍﺗﮋﯼ ‪ i‬ﺗﺮﺟﻴﺢ ﻧﺨﻮﺍﻫﺪ‬
‫ﺩﺍﺩ(‪ .‬ﺩﺭ ﭘﺎﻳﺎﻥ ﻣﯽﺗﻮﺍﻥ ﮔﻔﺖ ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ )ﺗﺮﮐﻴﺒﯽ(‪ ،‬ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﺁﻥ ﺗﻮﺯﻳﻊ ﺑﻪ‬
‫ﺻﻮﺭﺕ ﺣﺎﺻﻠﻀﺮﺑﯽ ﺍﺳﺖ‪.‬‬
‫‪ ٢,٢‬ﻣﺜﺎﻝﻫﺎﻳﻲ ﺍﺯ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‬
‫ﻳﮏ ﻣﺜﺎﻝ ﮐﻼﺳﻴﮏ ﺑﺮﺍﯼ ﺍﻳﻦ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ‪ ،‬ﺑﺎﺯﯼ ﭘﺮﻧﺪﻩ‪ ١٦‬ﺍﺳﺖ ﮐﻪ ﺷﺎﻣﻞ ﺩﻭ ﺑﺎﺯﻳﮑﻦ ﺍﺳﺖ‪ .‬ﻫﺮ ﮐﺪﺍﻡ ﺍﺯ ﺩﻭ ﺑﺎﺯﻳﮑﻦ ﺩﻭ‬
‫ﺍﺳﺘﺮﺍﺗﮋﯼ ﺩﺍﺭﻧﺪ‪ (۱ :‬ﻣﺒﺎﺭﺯﻩ )‪(D‬؛ ‪ (۲‬ﺍﻧﺼﺮﺍﻑ )‪ .(C‬ﺍﻟﺒﺘﻪ ﻣﯽﺗﻮﺍﻥ ﺍﻳﻦ ﺑﺎﺯﯼ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺩﻭ ﺭﺍﻧﻨﺪﻩ ﺑﺎ ﻋﺠﻠﻪ ﺩﺭ ﻧﻈﺮ‬
‫ﮔﺮﻓﺖ ﮐﻪ ﺩﺭ ﻳﮏ ﭼﻬﺎﺭﺭﺍﻩ ﻗﺮﺍﺭ ﺩﺍﺭﻧﺪ ﻭ ﺩﻭ ﺍﻧﺘﺨﺎﺏ ﺩﺍﺭﻧﺪ‪ (۱ :‬ﻋﺒﻮﺭ )‪(G‬؛ ‪ (۲‬ﺗﻮﻗﻒ )‪ .(S‬ﻣﻘﺪﺍﺭ ﻣﻨﻔﻌﺖ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺑﺎﺯﺍﯼ‬
‫ﺍﻧﺠﺎﻡ ﻫﺮ ﮐﺪﺍﻡ ﺍﺯ ﺍﺳﺘﺮﺍﺗﮋﯼﻫﺎ ﺩﺭ ﺟﺪﻭﻝ ﺯﻳﺮ ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪:‬‬
‫‪P2‬‬
‫)‪C (S‬‬
‫)‪D (G‬‬
‫)‪(5, 1‬‬
‫)‪(0, 0‬‬
‫)‪D (G‬‬
‫)‪(4, 4‬‬
‫)‪(1, 5‬‬
‫)‪C (S‬‬
‫‪P1‬‬
‫‪ ۵‬ﺗﻮﺯﻳﻊ ﺯﻳﺮ‪ ،‬ﺑﺮﺍﯼ ﺍﻳﻦ ﺑﺎﺯﯼ‪ ،‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﻣﺤﺴﻮﺏ ﻣﯽﺷﻮﻧﺪ‪:‬‬
‫ﺗﻮﺯﻳﻊﻫﺎﯼ ﺍﻭﻝ ﻭ ﺩﻭﻡ‪ ،‬ﺩﻭ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ ﺧﺎﻟﺺ ﺭﺍ ﻧﺸﺎﻥ ﻣﯽﺩﻫﻨﺪ‪ .‬ﺗﻮﺯﻳﻊ ﺳﻮﻡ ﻧﺸﺎﻥ ﺩﻫﻨﺪﻩ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻧﺶ‬
‫ﺗﺮﮐﻴﺒﯽ ﺍﺳﺖ ﮐﻪ ﻫﺮ ﺩﻭ ﺑﺎﺯﻳﮑﻦ ﺑﺎ ﺍﺳﺘﺮﺍﺗﮋﯼ ﺗﺮﮐﻴﺒﯽ }‪ {۱/۲ ،۱/۲‬ﻋﻤﻞ ﻣﯽﮐﻨﻨﺪ‪ .‬ﺗﻮﺯﻳﻊ ﭼﻬﺎﺭﻡ‪ ،‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺭﺍ‬
‫ﻧﺸﺎﻥ ﻣﯽﺩﻫﺪ ﮐﻪ ﻣﺜﻞ ﭘﺮﺗﺎﺏ ﺳﮑﻪ )ﻳﺎ ﭼﺮﺍﻍ ﺭﺍﻫﻨﻤﺎﻳﻲ ﺩﺭ ﺑﺎﺯﯼ ﻋﺒﻮﺭ ﻣﺎﺷﻴﻦﻫﺎ( ﻋﻤﻞ ﻣﯽﮐﻨﺪ ﻭ ﺑﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺳﺘﺮﺍﺗﮋﯼ‬
‫ﺭﺍ ﭘﻴﺸﻨﻬﺎﺩ ﻣﯽﮐﻨﺪ ﻭ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻫﻢ ﺗﻤﺎﻳﻠﯽ ﺑﻪ ﺳﺮﭘﻴﭽﯽ ﺍﺯ ﺍﻳﻦ ﺍﺳﺘﺮﺍﺗﮋﯼ ﻧﺨﻮﺍﻫﻨﺪ ﺩﺍﺷﺖ‪ .‬ﻣﺸﺨﺺ ﺍﺳﺖ ﺩﺭ ﺻﻮﺭﺕ‬
‫ﻣﺤﺎﺳﺒﻪ ﺍﻣﻴﺪ ﺭﻳﺎﺿﯽ ﻣﻨﻔﻌﺖ‪ ،‬ﺍﻳﻦ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻣﻨﺎﺳﺐﺗﺮ ﺍﺯ ﻧﻘﺎﻁ ﺗﻌﺎﺩﻝ ﻧﺶ ﻋﻤﻞ ﺧﻮﺍﻫﺪ ﻧﻤﻮﺩ‪ .‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‬
‫ﭘﻨﺠﻢ‪ ،‬ﺟﻤﻊ ﻣﻨﻔﻌﺖﻫﺎﯼ ﻣﻮﺭﺩ ﺍﻧﺘﻈﺎﺭ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺭﺍ ﺑﻴﺸﻴﻨﻪ ﻣﯽﮐﻨﺪ ﮐﻪ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺑﻴﺸﻴﻨﻪﺳﺎﺯﯼ ﺧﻄﯽ‪ ١٧‬ﺑﺪﺳﺖ ﺁﻣﺪﻩ‬
‫ﺍﺳﺖ‪.‬‬
‫‪ ٣‬ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ‬
‫ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﮐﻪ ﻣﯽﺗﻮﺍﻥ ﺑﻪ ﺁﻧﻬﺎ ﺑﺎﺯﯼﻫﺎﯼ ﭘﺮﺍﮐﻨﺪﻩ‪ ١٨‬ﻧﻴﺰ ﺍﻃﻼﻕ ﮐﺮﺩ‪ ،‬ﺑﺎﺯﯼﻫﺎﻳﻲ ﻫﺴﺘﻨﺪ ﮐﻪ ﺩﺭ ﺁﻥ ﺑﻌﻀﯽ ﺍﺯ ﻣﻘﺎﺩﻳﺮ‬
‫ﻣﻨﻔﻌﺖﻫﺎ ﺑﻪ ﺻﻮﺭﺕ ﺻﺮﻳﺢ ﺩﺍﺩﻩ ﻣﯽﺷﻮﺩ ﻭ ﺳﺎﻳﺮ ﻣﻮﺍﺭﺩ ﺻﻔﺮ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﻣﯽﺷﻮﺩ )ﺷﺒﻴﻪ ﺑﻪ ﻳﮏ ﻣﺎﺗﺮﻳﺲ ﭘﺮﺍﮐﻨﺪﻩ(‪.‬‬
‫ﺣﺎﻝ ﺍﮔﺮ ﺑﺨﻮﺍﻫﻴﻢ ﻳﮏ ﺑﺎﺯﯼ ﻓﺸﺮﺩﻩ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺭﺳﻤﯽ ﺗﻌﺮﻳﻒ ﻧﻤﺎﻳﻴﻢ‪ ،‬ﺑﺎﻳﺪ ﮔﻔﺖ‪ :‬ﻳﮏ ﺑﺎﺯﯼ ﻓﺸﺮﺩﻩ )‪G = (I, T, U‬‬
‫ﻣﺎﻧﻨﺪ ﺳﺎﻳﺮ ﻣﺴﺎﻳﻞ ﻣﺤﺎﺳﺒﺎﺗﯽ ﺑﺎ ﻳﮏ ﻣﺠﻤﻮﻋﻪ ﺍﺯ ﻭﺭﻭﺩﯼﻫﺎ )‪ (I‬ﺗﻌﺮﻳﻒ ﻣﯽﺷﻮﺩ‪ T .‬ﻭ ‪ U‬ﻫﻢ ﺩﻭ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺑﺎ ﭘﻴﭽﻴﺪﮔﯽ‬
‫ﺯﻣﺎﻧﯽ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﻫﺴﺘﻨﺪ‪ .‬ﺑﺎﺯﺍﯼ ﻫﺮ ‪ T(z) ، z ∈ I‬ﻳﮏ ﻧﻮﻉ‪ ١٩‬ﺑﺮﻣﯽﮔﺮﺩﺍﻧﺪ ﮐﻪ ﺑﺮﺍﯼ ‪ n‬ﺑﺎﺯﻳﮑﻦ‪ ،‬ﻳﮏ ‪ n‬ﺗﺎﻳﻲ ﻣﺮﺗﺐ‬
‫) ‪ (t1 , t 2 , ..., t n‬ﺭﺍ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ ﮐﻪ ﻧﺸﺎﻥﺩﻫﻨﺪﻩ ﻣﺠﻤﻮﻋﻪ ﺍﺳﺘﺮﺍﺗﮋﯼﻫﺎﯼ ﻣﻤﮑﻦ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺳﺖ‪ .‬ﺩﺭ ﺻﻮﺭﺗﯽ ﮐﻪ ‪ n‬ﻭ‬
‫‪ t p‬ﻫﺎ ﻧﺴﺒﺖ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﺑﺎ ‪ z‬ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ‪ ،‬ﮔﻔﺘﻪ ﻣﯽﺷﻮﺩ ﺑﺎﺯﯼ ﺩﺍﺭﺍﯼ ﻧﻮﻉ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﺍﺳﺖ‪ .‬ﺗﺎﺑﻊ )‪U(z, p, s‬‬
‫ﻣﻘﺪﺍﺭ ﻣﻨﻔﻌﺖ ﺑﺎﺯﻳﮑﻦ ‪ p‬ﺭﺍ ﺑﺎﺯﺍﯼ ﻭﺭﻭﺩﯼ ‪ z‬ﻭ ﺑﺎ ﺍﻧﺠﺎﻡ ﺍﺳﺘﺮﺍﺗﮋﯼ ‪ ( s = (s1 , s 2 , ..., s n ) ) s‬ﺗﻮﺳﻂ ﺑﺎﺯﻳﮑﻨﺎﻥ‪ ،‬ﻧﺸﺎﻥ‬
‫ﻣﯽﺩﻫﺪ‪.‬‬
‫ﺩﺭ ﺍﻳﻨﺠﺎ ﭼﻨﺪ ﻣﺜﺎﻝ ﺍﺯ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﺍﺭﺍﺋﻪ ﺧﻮﺍﻫﺪ ﺷﺪ‪ .‬ﺑﺎﺯﯼﻫﺎﯼ ﻣﺘﻘﺎﺭﻥ ﺍﻭﻟﻴﻦ ﺩﺳﺘﻪ ﺍﺯ ﺍﻳﻦ ﺑﺎﺯﯼﻫﺎ ﻫﺴﺘﻨﺪ ﮐﻪ‬
‫ﻧﺨﺴﺘﻴﻦ ﺑﺎﺭ ﺗﻮﺳﻂ ﻧﺶ ﻭ ﻓﻦ ﻧﻴﻮﻣﺎﻥ ﻣﻮﺭﺩ ﺑﺮﺭﺳﯽ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪﺍﻧﺪ ]‪ .[۲‬ﺩﺭ ﺍﻳﻦ ﺩﺳﺘﻪ‪ ،‬ﺑﺎﺯﻳﮑﻨﺎﻥ ﺷﺒﻴﻪ ﺑﻪ ﻫﻢ ﻭ ﻏﻴﺮ ﻗﺎﺑﻞ‬
‫ﺗﺸﺨﻴﺺﺍﻧﺪ ﻭ ﻣﻨﻔﻌﺖ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﻭﺍﺑﺴﺘﻪ ﺑﻪ ﺍﻧﺘﺨﺎﺏ ﺍﻭ ﻭ ﺗﻌﺪﺍﺩ ﺍﻧﺘﺨﺎﺏﻫﺎﻳﯽ ﺩﺍﺭﺩ ﮐﻪ ﺳﺎﻳﺮ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺯ ﺍﺳﺘﺮﺍﺗﮋﯼﻫﺎﯼ‬
‫ﻣﺘﻔﺎﻭﺕ ﺍﻧﺠﺎﻡ ﺩﺍﺩﻩﺍﻧﺪ‪ .‬ﺩﺳﺘﻪ ﺑﻌﺪﯼ‪ ،‬ﺑﺎﺯﯼﻫﺎﯼ ﺑﯽﻧﺎﻡ ﻫﺴﺘﻨﺪ ﮐﻪ ﻳﮏ ﺗﻌﻤﻴﻢ ﺑﺮﺍﯼ ﺑﺎﺯﯼﻫﺎﯼ ﻣﺘﻘﺎﺭﻥ ﻣﺤﺴﻮﺏ ﻣﯽﺷﻮﻧﺪ‬
‫ﮐﻪ ﺩﺭ ﺁﻥ‪ ،‬ﺗﺎﺑﻊ ﻣﻨﻔﻌﺖ ﺧﺎﺹ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺗﻌﺮﻳﻒ ﻣﯽﺷﻮﺩ‪ .‬ﺩﺳﺘﻪ ﺳﻮﻡ‪ ،‬ﺑﺎﺯﯼﻫﺎﯼ ﮔﺮﺍﻓﻴﮑﯽ ﻫﺴﺘﻨﺪ ﮐﻪ ﺩﺭ ﺁﻥ ﻳﮏ ﺷﺒﮑﻪ‬
‫ﻣﺘﺼﻞ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺩﺍﺭﻳﻢ ﻭ ﻣﻨﻔﻌﺖ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺑﻪ ﺍﻧﺘﺨﺎﺏ ﺧﻮﺩ ﻭ ﻫﻤﺴﺎﻳﮕﺎﻧﺶ ﻭﺍﺑﺴﺘﻪ ﺍﺳﺖ‪ .‬ﺑﺎﺯﯼﻫﺎﯼ ﭼﻨﺪ ﻣﺎﺗﺮﻳﺴﯽ‬
‫ﺩﺳﺘﻪ ﺑﻌﺪﯼ ﺍﺯ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﻫﺴﺘﻨﺪ ﮐﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﻳﮏ ﺑﺎﺯﯼ ﺩﻭ ﻧﻔﺮﻩ ﺑﺎ ﺳﺎﻳﺮ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﻧﺠﺎﻡ ﻣﯽﺩﻫﺪ ﻭ ﻣﻨﻔﻌﺖ‬
‫ﺑﺎﺯﻳﮑﻦ ﺑﺮﺍﺑﺮ ﻣﻨﻔﻌﺖ ﻣﺠﻤﻮﻉ ﺑﺎﺯﯼﻫﺎﯼ ﺍﻧﺠﺎﻡ ﺷﺪﻩ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ .‬ﺑﺎﺯﯼﻫﺎﯼ ﺩﻳﮕﺮﯼ ﻧﻈﻴﺮ ﺑﺎﺯﻱﻫﺎﯼ ﺍﺑﺮﮔﺮﺍﻓﻴﮑﯽ‪،٢٠‬‬
‫ﺑﺎﺯﯼﻫﺎﯼ ﺍﺯﺩﺣﺎﻡ‪ ٢١‬ﻭ ﺑﺎﺯﯼﻫﺎﯼ ﺯﻣﺎﻥﺑﻨﺪﯼ ﻣﺜﺎﻝﻫﺎﯼ ﺩﻳﮕﺮﯼ ﺍﺯ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﻫﺴﺘﻨﺪ‪.‬‬
‫‪ ٤‬ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﺑﺘﺪﺍﻳﻲ ﭘﻴﺪﺍ ﻧﻤﻮﺩﻥ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‬
‫ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺭﻭﯼ ﺩﻭﮔﺎﻥ )‪ (D‬ﺍﺯ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﻋﻤﻞ ﻣﯽﮐﻨﺪ ﮐﻪ ﺷﺎﻣﻞ ﺗﻌﺪﺍﺩ ﺯﻳﺎﺩﯼ ﻣﺘﻐﻴﺮ ﺑﺎ ﺩﺭﺟﻪ‬
‫ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﻭ ﺗﻌﺪﺍﺩ ﺯﻳﺎﺩﯼ ﻣﺤﺪﻭﺩﻳﺖ ﺑﺎ ﺩﺭﺟﻪ ﻧﻤﺎﻳﻲ ﺍﺳﺖ‪ .‬ﺑﺎ ﻭﺟﻮﺩ ﺍﻳﻨﮑﻪ ﺩﺭ ﺍﺛﺒﺎﺕ ﻭﺟﻮﺩ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‬
‫ﻧﺘﻴﺠﻪﮔﻴﺮﯼ ﻣﯽﺷﻮﺩ ﮐﻪ ﺣﻞ )‪ (D‬ﻧﺎﻣﻤﮑﻦ ﺍﺳﺖ‪ ،‬ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﻟﻴﭙﺴﻮﻳﺪ ]‪ [۵‬ﺭﻭﯼ ﺁﻥ ﺍﻋﻤﺎﻝ ﻣﯽﺷﻮﺩ‪ .‬ﺣﺎﻝ ﺍﺯ ﻣﺤﺎﺳﺒﺎﺕ‬
‫ﺯﻧﺠﻴﺮﻩ ﻣﺎﺭﮐﻮﻑ ﺩﺭ ﻫﺮ ﻣﺮﺣﻠﻪ ﺍﺯ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﻟﻴﭙﺴﻮﻳﺪ ﺍﺳﺘﻔﺎﺩﻩ ﻣﯽﺷﻮﺩ ﺗﺎ ﻳﮏ ﺗﺮﮐﻴﺐ ﻣﺤﺪﺏ ﺍﺯ ﻣﺤﺪﻭﺩﻳﺖﻫﺎﯼ )‪ (D‬ﮐﻪ‬
‫ﻣﺘﻨﺎﻗﺾ ﻫﺴﺘﻨﺪ‪ ،‬ﭘﻴﺪﺍ ﺷﻮﺩ‪ .‬ﺩﺭ ﭘﺎﻳﺎﻥ ﺍﻟﮕﻮﺭﻳﺘﻢ‪ ،‬ﻳﮏ ﺗﻌﺪﺍﺩ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺍﺯ ﺍﻳﻦ ﺗﺮﮐﻴﺐﻫﺎ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ ﮐﻪ ﺣﻞﺷﺎﻥ‬
‫ﻧﺎﻣﻤﮑﻦ ﺍﺳﺖ ﻭ ﺁﻧﻬﺎ ﺭﺍ )‪ (D′‬ﻣﯽﻧﺎﻣﻴﻢ‪ .‬ﺣﻞ ﮐﺮﺩﻥ ﺩﻭﮔﺎﻥ ﺍﻳﻦ ﺑﺮﻧﺎﻣﻪ ﺧﻄﯽ ﺟﺪﻳﺪ )ﺗﺤﺖ ﻋﻨﻮﺍﻥ )‪ ،( (P ′‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ‬
‫ﻫﻤﺒﺴﺘﻪ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ "ﺗﺮﮐﻴﺐ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺍﺯ ﺣﺎﺻﻠﻀﺮﺏﻫﺎ"‪ ٢٢‬ﺑﺪﺳﺖ ﻣﯽﺩﻫﺪ‪.‬‬
‫ﺍﻟﺒﺘﻪ ﺑﺮﺍﯼ ﺣﻞ ﮐﺮﺩﻥ ‪ ، P′‬ﺍﺣﺘﻴﺎﺝ ﺑﻪ ﻳﮏ ﺗﮑﻨﻴﮏ ﺧﺎﺹ ﻣﺴﺎﻟﻪ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﮐﻪ ﺑﺮﺍﯼ ﺗﻤﺎﻣﯽ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﭼﻨﻴﻦ‬
‫ﺗﮑﻨﻴﮏﻫﺎﻳﯽ ﮐﻪ ﺩﺭ ﺯﻣﺎﻥ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﻗﺎﺑﻞ ﺍﻋﻤﺎﻝ ﻫﺴﺘﻨﺪ‪ ،‬ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪ ،‬ﻣﺜﻼ ﻣﯽﺗﻮﺍﻥ ﺍﺯ ﺗﮑﻨﻴﮏ ﺧﻄﯽ ﺑﻮﺩﻥ ﺍﻣﻴﺪ‪ ٢٣‬ﺩﺭ‬
‫ﺑﺎﺯﯼﻫﺎﯼ ﭼﻨﺪﻣﺎﺗﺮﻳﺴﯽ ﻭ ﺍﺑﺮﮔﺮﺍﻓﻴﮑﯽ‪ ،‬ﺍﺯ ﺗﮑﻨﻴﮏ ﺑﺮﻧﺎﻣﻪﺳﺎﺯﯼ ﭘﻮﻳﺎ ﺩﺭ ﺑﺎﺯﯼﻫﺎﯼ ﺯﻣﺎﻥﺑﻨﺪﯼ ﻭ ﺍﺯﺩﺣﺎﻡ ﻳﺎ ﺍﺯ ﺗﮑﻨﻴﮏ‬
‫ﺗﻌﺮﻳﻒ ﺍﻣﻴﺪ ﺩﺭ ﺩﺍﻣﻨﻪﻫﺎﯼ ﮐﻮﭼﮏ ﺑﺮﺍﯼ ﺑﺎﺯﯼﻫﺎﯼ ﮔﺮﺍﻓﻴﮑﯽ ﻭ ﺍﺑﺮﮔﺮﺍﻓﻴﮑﯽ ﺍﺳﺘﻔﺎﺩﻩ ﻧﻤﻮﺩ‪ .‬ﺩﺭ ﺍﺩﺍﻣﻪ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺑﺎ ﺟﺰﺋﻴﺎﺕ‬
‫ﺑﻴﺸﺘﺮﯼ ﻣﻮﺭﺩ ﺑﺮﺭﺳﯽ ﻗﺮﺍﺭ ﺧﻮﺍﻫﺪ ﮔﺮﻓﺖ‪:‬‬
‫‪ ١,٤‬ﺍﺛﺒﺎﺕ ﻭﺟﻮﺩ‬
‫ﺩﺭ ﺍﻳﻨﺠﺎ‪ ،‬ﺍﺯ ﺗﻐﻴﻴﺮﺍﺗﯽ ﮐﻪ ﺩﺭ ﺍﺛﺒﺎﺕ ﻭﺟﻮﺩ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ]‪ [۳‬ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ‪ ،‬ﺍﺳﺘﻔﺎﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪.‬‬
‫ﻗﻀﻴﻪ ‪ :۱‬ﻫﺮ ﺑﺎﺯﯼ ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺩﺍﺭﺩ‪.‬‬
‫ﺍﺛﺒﺎﺕ‪ :‬ﺑﺮﻧﺎﻣﻪ ﺧﻄﯽ ﺯﻳﺮ ﺭﺍ ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﺪ‪:‬‬
‫ﻣﺤﺪﻭﺩﻳﺖﻫﺎﯼ ﺑﻴﺎﻥ ﺷﺪﻩ ﺩﺭ )‪ (P‬ﻫﻤﺎﻥ ﻣﺤﺪﻭﺩﻳﺖﻫﺎﯼ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺍﺳﺖ ﮐﻪ ﺑﺮﺍﯼ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﻭ ﻫﺮ ﺯﻭﺝ‬
‫ﺍﺳﺘﺮﺍﺗﮋﯼﻫﺎ ﺑﺮﻗﺮﺍﺭ ﺍﺳﺖ‪ .‬ﺑﺮﻧﺎﻣﻪ )‪ (P‬ﻳﺎ ﺧﻴﻠﯽ ﺳﺎﺩﻩ ﺍﺳﺖ )ﺑﺎ ﻧﻘﻄﻪ ﺑﻴﺸﻴﻨﻪ ‪ (۰‬ﻭ ﻳﺎ ﻧﺎﻣﺤﺪﻭﺩ ﺍﺳﺖ ﮐﻪ ﺍﻳﻦ ﺣﺎﻟﺖ ﺩﻗﻴﻘﺎ‬
‫ﻭﻗﺘﯽ ﺍﺗﻔﺎﻕ ﻣﯽﺍﻓﺘﺪ ﮐﻪ ﺑﺎﺯﯼ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ‪ ،‬ﻃﺒﻖ ﺩﻭﮔﺎﻧﮕﯽ‪ ،‬ﺑﺮﺍﯼ ﺍﺛﺒﺎﺕ ﻗﻀﻴﻪ ﮐﺎﻓﯽ ﺍﺳﺖ‬
‫ﻧﺸﺎﻥ ﺩﻫﻴﻢ ﮐﻪ ﺩﻭﮔﺎﻥ )‪(P‬‬
‫ﻫﻤﻴﺸﻪ ﻧﺎﻣﻤﮑﻦ ﺍﺳﺖ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﻟﻢ ﺯﻳﺮ ﻣﻄﺮﺡ ﻣﯽﺷﻮﺩ‪:‬‬
‫ﻟﻢ ‪ :۱‬ﺑﺎﺯﺍﯼ ﻫﺮ ‪ ، y ≥ 0‬ﻳﮏ ﺗﻮﺯﻳﻊ ﺣﺎﺻﻠﻀﺮﺑﯽ ‪ x‬ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﺑﻪ ﻧﺤﻮﯼ ﮐﻪ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪. xU T y = 0 :‬‬
‫ﺑﺎ ﺩﻗﺖ ﻣﯽﺗﻮﺍﻥ ﻓﻬﻤﻴﺪ ﮐﻪ ‪ ، xU T y‬ﻳﮏ ﺗﺮﮐﻴﺐ ﻣﺤﺪﺏ ﺍﺯ ﻣﺤﺪﻭﺩﻳﺖﻫﺎﯼ ﺳﻤﺖ ﭼﭗ ﺑﺮﻧﺎﻣﻪ )‪ (D‬ﺍﺳﺖ ﻭ‬
‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻫﺮ ‪ y‬ﻣﻤﮑﻦ‪ ،‬ﺑﺎﻳﺪ ﺩﺍﺭﺍﯼ ﻣﻘﺪﺍﺭﯼ ﻣﻨﻔﯽ ﺑﺎﺷﺪ‪ ،‬ﭘﺲ ﻣﯽﺗﻮﺍﻥ ﮔﻔﺖ ﮐﻪ ﺣﻞ )‪ (D‬ﻧﺎﻣﻤﮑﻦ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ ،‬ﺯﻳﺮﺍ ﻳﮑﯽ‬
‫ﺍﺯ ﻣﺤﺪﻭﺩﻳﺖﻫﺎﯼ ﺁﻥ ﺑﻪ ﻣﺜﺒﺖ ﺑﻮﺩﻥ ‪ y‬ﺍﺷﺎﺭﻩ ﺩﺍﺭﺩ‪ .‬ﺍﻟﺒﺘﻪ ﺑﺮﺍﯼ ﺍﺛﺒﺎﺕ ﻟﻢ ‪ ۱‬ﺍﺯ ﻣﺤﺎﺳﺒﺎﺕ ﻭﺿﻌﻴﺖ ﭘﺎﻳﺪﺍﺭ ﺩﺭ ﺯﻧﺠﻴﺮﻩ‬
‫ﻣﺎﺭﮐﻮﻑ ﺍﺳﺘﻔﺎﺩﻩ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﺑﺪﻟﻴﻞ ﻣﺤﺪﻭﺩﻳﺖ ﻓﻀﺎ ﺍﺯ ﺑﻴﺎﻥ ﺁﻥ ﺻﺮﻓﻨﻈﺮ ﻣﯽﺷﻮﺩ )ﺑﻪ ]‪ [۱‬ﻣﺮﺍﺟﻌﻪ ﺷﻮﺩ(‪.‬‬
‫‪ ٢,٤‬ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﻟﻴﭙﺴﻮﻳﺪ‬
‫ﺣﺎﻝ ﺑﺎﻳﺪ ﺍﻳﻦ ﺍﺛﺒﺎﺕ ﻭﺟﻮﺩ ﺭﺍ ﺑﻪ ﺳﻤﺖ ﻳﮏ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺑﺎ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﻣﺘﻤﺎﻳﻞ ﻧﻤﻮﺩ‪ .‬ﺍﻳﺪﻩ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ‬
‫ﺍﻟﮕﻮﺭﻳﺘﻢ ﺍﻟﻴﭙﺴﻮﻳﺪ ﺑﺮ ﺩﻭﮔﺎﻥ )‪ (D‬ﮐﻪ ﻣﺸﺨﺺ ﺷﺪ ﻧﺎﻣﻤﮑﻦ ﺍﺳﺖ‪ ،‬ﺍﻋﻤﺎﻝ ﺷﻮﺩ‪ .‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺧﺼﻮﺻﻴﺎﺕ )‪ (D‬ﮐﻪ ﺩﺍﺭﺍﯼ‬
‫ﺗﻌﺪﺍﺩ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﻣﺘﻐﻴﺮ ﻭ ﺗﻌﺪﺍﺩ ﻧﻤﺎﻳﻲ ﻣﺤﺪﻭﺩﻳﺖ ﺍﺳﺖ )ﺑﺮﻧﺎﻣﻪ )‪ (P‬ﺍﺯ ﺍﻳﻦ ﺟﻬﺖ ﺑﺮﻋﮑﺲ )‪ (D‬ﺍﺳﺖ(‪ ،‬ﺍﻟﮕﻮﺭﻳﺘﻢ‬
‫ﺍﻟﻴﭙﺴﻮﻳﺪ ﺑﺮﺍﯼ ﺍﻋﻤﺎﻝ ﺭﻭﯼ )‪ (D‬ﻣﻨﺎﺳﺐ ﺍﺳﺖ‪ .‬ﺩﺭ ﻫﺮ ﻣﺮﺣﻠﻪ ‪ ،i‬ﻳﮏ ﺭﺍﻩ ﺣﻞ ﮐﺎﻧﺪﻳﺪ ‪ yi‬ﻭﺟﻮﺩ ﺩﺍﺭﺩ؛ ﺍﺯ ﻟﻢ ‪ ۱‬ﺍﺳﺘﻔﺎﺩﻩ‬
‫ﻣﯽﺷﻮﺩ ﺗﺎ ‪ xi‬ﯼ ﺑﺪﺳﺖ ﺁﻳﺪ ﮐﻪ ﻧﺎﻣﺴﺎﻭﯼ ‪ xiU T y ≤ − 1‬ﺭﺍ ﻧﻘﺾ ﮐﻨﺪ‪ .‬ﭘﺲ ﺍﺯ ﺁﻥ ﺍﻟﮕﻮﺭﻳﺘﻢ ﻭﺍﺭﺩ ﻣﺮﺣﻠﻪ ﺑﻌﺪﯼ‬
‫ﻣﯽﺷﻮﺩ‪ .‬ﭘﺲ ﺍﺯ ﺍﺗﻤﺎﻡ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺩﺭ ﮔﺎﻡ ‪ L‬ﺍﻡ )‪ L‬ﺑﻪ ﺻﻮﺭﺕ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﺍﺳﺖ(‪ ،‬ﺗﻌﺪﺍﺩ ‪ L‬ﺗﻮﺯﻳﻊ ﺣﺎﺻﻠﻀﺮﺑﯽ‬
‫‪ x1 , x2 , ..., x L‬ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﮐﻪ ﺑﺎﺯﺍﯼ ‪ i ≤ L‬ﻧﺎﻣﺴﺎﻭﯼ ‪ [xiU T ] y ≤ − 1‬ﺗﻮﺳﻂ ‪ yi‬ﻧﻘﺾ ﻣﯽﺷﻮﺩ‪ .‬ﺍﻳﻦ ﻧﺘﻴﺠﻪ ﺑﻪ ﺍﻳﻦ‬
‫ﻣﻌﻨﯽ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﻧﺎﻣﺴﺎﻭﯼ ‪ X) ، [XU T ] y ≤ − 1‬ﻣﺎﺗﺮﻳﺴﯽ ﺍﺳﺖ ﮐﻪ ﺭﺩﻳﻒﻫﺎﯼ ﺁﻥ ‪ xi‬ﻫﺎ ﻫﺴﺘﻨﺪ(‪ ،‬ﺧﻮﺩ ‪ X‬ﻳﮏ‬
‫ﺑﺮﻧﺎﻣﻪ ﺧﻄﯽ ﻧﺎﻣﻤﮑﻦ ﺍﺳﺖ‪.‬‬
‫ﺑﺎﺗﻮﺟﻪ ﺑﻪ ﻧﺎﻣﻤﮑﻦ ﺑﻮﺩﻥ ﺑﺮﻧﺎﻣﻪ ‪ ،X‬ﻣﯽﺗﻮﺍﻥ ﻧﺘﻴﺠﻪ ﮔﺮﻓﺖ ﮐﻪ ﺩﻭﮔﺎﻥ ﺁﻥ ﻳﻌﻨﯽ ﺑﺮﻧﺎﻣﻪ‬
‫‪[UX ] α ≥ 0 , α ≥ 0‬‬
‫‪T‬‬
‫ﻧﺎﻣﺤﺪﻭﺩ ﺍﺳﺖ‪ .‬ﭼﻨﻴﻦ ﺑﺮﺩﺍﺭ ﻏﻴﺮ ﺻﻔﺮ ‪ ، α‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﻣﺪﻧﻈﺮ ﺭﺍ ﻓﺮﺍﻫﻢ ﻣﯽﮐﻨﺪ ﮐﻪ ﺩﺭ ﻭﺍﻗﻊ ﻳﮏ ﺗﺮﮐﻴﺐ‬
‫ﻣﺤﺪﺏ ﺍﺯ ‪ xi‬ﻫﺎﻳﻲ ﺍﺳﺖ ﮐﻪ )‪ (P‬ﺭﺍ ﺍﺭﺿﺎ ﻣﯽﮐﻨﻨﺪ‪.‬‬
‫‪ ٣,٤‬ﺗﺮﮐﻴﺒﺎﺕ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﺍﺯ ﺣﺎﺻﻠﻀﺮﺏﻫﺎ‬
‫ﺩﺭ ﺍﻳﻦ ﺯﻳﺮﺑﺨﺶ‪ ،‬ﻧﺘﻴﺠﻪ ﺍﺻﻠﯽ ﺑﺨﺶ ﭼﻬﺎﺭﻡ ﺍﺭﺍﺋﻪ ﻣﯽﺷﻮﺩ ﮐﻪ ﻳﮏ ﻗﻀﻴﻪ ﺗﻘﻮﻳﺖ ﺷﺪﻩ ﺑﺮﺍﯼ ﻗﻀﻴﻪ ﻭﺟﻮﺩ )ﻗﻀﻴﻪ ‪(۱‬‬
‫ﺍﺳﺖ‪ .‬ﻧﺘﺎﻳﺞ ﺍﻟﮕﻮﺭﻳﺘﻤﯽ ﺁﻥ ﺩﺭ ﺑﺨﺶ ﺑﻌﺪ ﺍﺭﺍﺋﻪ ﺧﻮﺍﻫﺪ ﺷﺪ‪.‬‬
‫ﻗﻀﻴﻪ ‪ :۲‬ﻫﺮ ﺑﺎﺯﯼ ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺩﺍﺭﺩ ﮐﻪ ﺣﺎﺻﻞ ﺗﺮﮐﻴﺐ ﻣﺤﺪﺏ ﺍﺯ ﺗﻮﺯﻳﻊﻫﺎﯼ ﺣﺎﺻﻠﻀﺮﺑﯽ ﺍﺳﺖ ﮐﻪ ﺗﻌﺪﺍﺩ‬
‫ﺁﻥ ﺑﻪ ﺻﻮﺭﺕ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺍﺳﺖ‪.‬‬
‫ﺑﺮﺍﯼ ﻣﺸﺎﻫﺪﻩ ﺍﺛﺒﺎﺕ ﺑﻪ ]‪ [۲ ،۱‬ﺭﺟﻮﻉ ﺷﻮﺩ‪.‬‬
‫‪ ٥‬ﻣﺤﺎﺳﺒﻪ ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‬
‫ﻳﮏ ﺑﺎﺯﯼ ﻓﺸﺮﺩﻩ ‪ G‬ﺭﺍ ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﺪ‪ G .‬ﺩﺍﺭﺍﯼ ﺧﺎﺻﻴﺖ ﺍﻣﻴﺪ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ‪ ٢٤‬ﺍﺳﺖ‪ ،‬ﺍﮔﺮ ﻳﮏ ﺍﻟﮕﻮﺭﻳﺘﻢ ‪ ε‬ﺑﺎ‬
‫ﭘﻴﭽﻴﺪﮔﯽ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﻭﺟﻮﺩ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ ﮐﻪ ﺑﺎ ﺩﺍﺩﻥ ‪ p ≤ n ، z ∈ I‬ﻭ ﻳﮏ ﺗﻮﺯﻳﻊ ﺣﺎﺻﻠﻀﺮﺑﯽ } ‪ ، {x s : s ∈ S‬ﺑﺘﻮﺍﻧﺪ‬
‫ﺩﺭ ﺯﻣﺎﻥ ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﺍﻣﻴﺪ ﻣﻨﻔﻌﺖ ‪ u p s‬ﺭﺍ ﺑﺎ ﺩﺍﺩﻥ ﺗﻮﺯﻳﻊ ﺣﺎﺻﻠﻀﺮﺑﯽ ﻣﻮﺭﺩ ﻧﻈﺮ ﻣﺤﺎﺳﺒﻪ ﮐﻨﺪ‪:‬‬
‫ﻗﻀﻴﻪ ‪ :۳‬ﻓﺮﺽ ﮐﻨﻴﺪ ‪ G‬ﻳﮏ ﺑﺎﺯﯼ ﻓﺸﺮﺩﻩ ﺑﺎﺷﺪ ﮐﻪ ﺩﺍﺭﺍﯼ ﻧﻮﻉ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺑﺎﺷﺪ ﻭ ﺧﺎﺻﻴﺖ ﺍﻣﻴﺪ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺩﺍﺷﺘﻪ‬
‫ﺑﺎﺷﺪ‪ .‬ﺁﻧﮕﺎﻩ ﺑﺎﺯﯼ ﺩﺍﺭﺍﯼ ﻳﮏ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﺎ ﭘﻴﭽﻴﺪﮔﯽ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬
‫ﺑﺮﺍﯼ ﻣﺸﺎﻫﺪﻩ ﺍﺛﺒﺎﺕ ﺑﻪ ]‪ [۱‬ﺭﺟﻮﻉ ﺷﻮﺩ‪.‬‬
‫ﻣﯽﺗﻮﺍﻥ ﻧﺸﺎﻥ ﺩﺍﺩ ﮐﻪ ﺗﻤﺎﻣﯽ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﮐﻪ ﻧﻮﻉ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺩﺍﺭﻧﺪ‪ ،‬ﺩﺍﺭﺍﯼ ﺧﺎﺻﻴﺖ ﺍﻣﻴﺪ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ‬
‫ﻫﺴﺘﻨﺪ ]‪ .[۱‬ﺩﺭ ﺑﺨﺶ ﺑﻌﺪ ﺧﻮﺍﻫﻴﻢ ﺩﻳﺪ ﮐﻪ ﺑﻬﻴﻨﻪﺳﺎﺯﯼ ﺩﺭ ﻧﻘﺎﻁ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ‪ ،‬ﺩﺭ ﺑﺴﻴﺎﺭﯼ ﺍﺯ ﺍﻳﻦ ﺑﺎﺯﯼﻫﺎ ﻣﺴﺎﻟﻪﺍﯼ‬
‫‪ NP-hard‬ﺍﺳﺖ‪.‬‬
‫‪ ٦‬ﭘﻴﺪﺍ ﮐﺮﺩﻥ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ‬
‫ﺩﺭ ﺑﺨﺶ ﻗﺒﻞ ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪ ﮐﻪ ﭼﮕﻮﻧﻪ ﻣﯽﺗﻮﺍﻥ ﺑﻪ ﺻﻮﺭﺕ ﮐﺎﺭﺍ ﺑﻌﻀﯽ ﺍﺯ ﻧﻘﺎﻁ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺭﺍ ﺑﺮﺍﯼ ﺑﺎﺯﯼﻫﺎﯼ‬
‫ﻓﺸﺮﺩﻩ ﺑﺪﺳﺖ ﺁﻭﺭﺩ‪ .‬ﺣﺎﻝ ﻳﮏ ﺳﻮﺍﻝ ﻣﻬﻢﺗﺮ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﺁﻳﺎ ﻣﯽﺗﻮﺍﻥ ﺩﺭ ﺯﻣﺎﻥ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ‪ ،‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ‬
‫ﺭﺍ ﺑﺮﺍﯼ ﺍﻳﻦ ﺑﺎﺯﯼﻫﺎ ﺑﺪﺳﺖ ﺁﻭﺭﺩ ﻳﺎ ﺧﻴﺮ‪ .‬ﺩﺭ ﺍﺩﺍﻣﻪ ﺍﻳﻦ ﻣﺴﺎﻟﻪ ﻣﻮﺭﺩ ﺑﺮﺭﺳﯽ ﻗﺮﺍﺭ ﺧﻮﺍﻫﺪ ﮔﺮﻓﺖ‪.‬‬
‫‪ ١,٦‬ﺗﻌﺮﻳﻒ ﻣﺴﺎﻟﻪ‬
‫ﺣﺎﻝ ﺍﮔﺮ ﺑﺨﻮﺍﻫﻴﻢ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ ﺭﺍ ﺑﺪﺳﺖ ﺁﻭﺭﻳﻢ‪ ،‬ﺑﺎﻳﺪ ﺑﺘﻮﺍﻧﻴﻢ ﺟﻤﻊ ﻣﻨﻔﻌﺖﻫﺎﯼ ﻣﻮﺭﺩ ﺍﻧﺘﻈﺎﺭ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺭﺍ‬
‫ﺑﻴﺸﻴﻨﻪ ﮐﻨﻴﻢ‪ ،‬ﻳﻌﻨﯽ ﺑﺎﻳﺪ ﺑﻪ ﺑﺮﻧﺎﻣﻪ ﺧﻄﯽ ﺍﻭﻟﻴﻪ )‪ ،(P‬ﺭﺍﺑﻄﻪ‬
‫ﻧﺸﺎﻥﺩﻫﻨﺪﻩ‬
‫‪p‬‬
‫‪us‬‬
‫‪p‬‬
‫∑‬
‫ﺍﺳﺖ ﻭ ﻣﺤﺪﻭﺩﻳﺖ ‪= 1‬‬
‫‪s‬‬
‫‪∑x‬‬
‫‪s‬‬
‫‪max ∑ u s xs‬‬
‫‪s∈S‬‬
‫ﺍﺿﺎﻓﻪ ﺷﻮﺩ‪ ،‬ﮐﻪ ﺩﺭ ﺍﻳﻦ ﺭﺍﺑﻄﻪ‪u s ،‬‬
‫ﻫﻢ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪ .‬ﻣﺤﺪﻭﺩﻳﺖ ﺟﺪﻳﺪ ﺑﺎﻋﺚ ﻣﯽﺷﻮﺩ ﻣﺘﻐﻴﺮ‬
‫ﺟﺪﻳﺪ ‪ z‬ﺩﺭ ﺩﻭﮔﺎﻥ ﻭﺍﺭﺩ ﺷﻮﺩ ﻭ ﻣﺤﺪﻭﺩﻳﺖ ﺟﺪﻳﺪ ﺯﻳﺮ ﺑﻪ ﺑﺮﻧﺎﻣﻪ ﺩﻭﮔﺎﻥ ﺍﺿﺎﻓﻪ ﺧﻮﺍﻫﺪ ﺷﺪ‪:‬‬
‫)‪(1‬‬
‫ﺩﺭ ﺍﻳﻦ ﺭﺍﺑﻄﻪ‪ U s ،‬ﻧﺸﺎﻥﻫﻨﺪﻩ ﺳﺘﻮﻥ ﻣﺎﺗﺮﻳﺲ ‪ U‬ﺍﺳﺖ ﮐﻪ ﻣﺘﻨﺎﻇﺮ ﺑﺎ ﺍﺳﺘﺮﺍﺗﮋﯼ ‪ s‬ﺍﺳﺖ‪ .‬ﺩﺭ ﺑﺮﻧﺎﻣﻪ ﺧﻄﯽ ﺍﻭﻟﻴﻪ ﺗﻨﻬﺎ‬
‫ﮐﺎﻓﯽ ﺑﻮﺩ ﺑﺮﺩﺍﺭ ‪ x‬ﻏﻴﺮ ﺻﻔﺮ ﻭ ﻧﺎﻣﻨﻔﯽ ﺑﺪﺳﺖ ﺁﻳﺪ ﮐﻪ ‪ xU T y = 0‬ﺭﺍ ﺍﺭﺿﺎ ﻧﻤﺎﻳﺪ‪ .‬ﺍﻣﺎ ﺩﺭ ﺍﻳﻦ ﺑﺮﻧﺎﻣﻪ ﺟﺪﻳﺪ‪ ،‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ‬
‫ﺳﻤﺖ ﺭﺍﺳﺖ ﺭﺍﺑﻄﻪ )‪ (۱‬ﮐﻪ ﺩﻭ ﻣﺘﻐﻴﺮ ‪ y‬ﻭ ‪ z‬ﻭﺟﻮﺩ ﺩﺍﺭﻧﺪ‪ ،‬ﺑﺮﺩﺍﺭ ‪ x‬ﻣﯽﺗﻮﺍﻧﺪ ﻣﻨﻔﯽ ﻳﺎ ﻣﺜﺒﺖ ﺑﺎﺷﺪ ﻭ ﭼﻨﻴﻦ ‪ x‬ﯼ ﺳﻮﺩﻣﻨﺪ‬
‫ﻧﺨﻮﺍﻫﺪ ﺑﻮﺩ‪ .‬ﺍﻳﻦ ﺑﺎﻋﺚ ﻣﯽﺷﻮﺩ ﺑﺮﺍﯼ ﺑﺴﻴﺎﺭﯼ ﺍﺯ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ‪ ،‬ﻣﺴﺎﻟﻪ ‪ NP-hard‬ﺑﺎﺷﺪ‪.‬‬
‫ﺍﻟﺒﺘﻪ ﺩﺭ ﻣﺜﺎﻝﻫﺎﯼ ﻣﻬﻤﯽ ﻧﻈﻴﺮ ﺑﻌﻀﯽ ﺍﺯ ﺑﺎﺯﯼﻫﺎﯼ ﮔﺮﺍﻓﻴﮑﯽ ﻭ ﺑﺎﺯﯼﻫﺎﯼ ﻣﺘﻘﺎﺭﻥ‪ ،‬ﻣﺤﺪﻭﺩﻳﺖﻫﺎﯼ ﻧﻤﺎﻳﻲ ﺭﺍ ﻣﯽﺗﻮﺍﻥ‬
‫ﺑﻪ ﺗﻌﺪﺍﺩ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺍﺯ "ﮐﻼﺱﻫﺎﯼ ﻣﺴﺎﻭﯼ"‪ ٢٥‬ﮐﻪ ﺧﺎﺹ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺳﺖ‪ ،‬ﺗﺒﺪﻳﻞ ﻧﻤﻮﺩ‪ .‬ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ ﻣﯽﺗﻮﺍﻥ ﻧﻘﻄﻪ‬
‫ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ ﺭﺍ ﺩﺭ ﺯﻣﺎﻥ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺑﺪﺳﺖ ﺁﻭﺭﺩ‪ .‬ﺩﺭ ﺍﺩﺍﻣﻪ ﺑﻪ ﺫﮐﺮ ﺩﻭ ﻗﻀﻴﻪ ﻣﻬﻢ ﺩﺭ ﻣﻮﺭﺩ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ‬
‫ﭘﻴﺪﺍ ﮐﺮﺩﻥ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ ﺑﺴﻨﺪﻩ ﺧﻮﺍﻫﻴﻢ ﮐﺮﺩ‪.‬‬
‫ﻗﻀﻴﻪ ‪ :۴‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ ﺑﺮﺍﯼ ﺑﺎﺯﯼﻫﺎﯼ ﺑﯽﻧﺎﻡ ﻭ ﺑﺎﺯﯼﻫﺎﯼ ﮔﺮﺍﻓﻴﮑﯽ ﺑﺎ ﻋﺮﺽ ﺩﺭﺧﺘﯽ ﻣﺤﺪﻭﺩ‪ ،٢٦‬ﺩﺭ ﺯﻣﺎﻥ‬
‫ﭼﻨﺪﺟﻤﻠﻪﺍﯼ ﻗﺎﺑﻞ ﻣﺤﺎﺳﺒﻪ ﺍﺳﺖ ]‪.[۱‬‬
‫ﻗﻀﻴﻪ ‪ :۵‬ﺩﺭ ﻣﻮﺭﺩ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﻣﻘﺎﺑﻞ‪ ،‬ﻣﺤﺎﺳﺒﻪ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ ﺩﺍﺭﺍﯼ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ ‪ NP-hard‬ﺍﺳﺖ‪:‬‬
‫ﺑﺎﺯﯼﻫﺎﯼ ﮔﺮﺍﻓﻴﮑﯽ ﺩﻭﻗﺴﻤﺘﯽ‪ ،٢٧‬ﭼﻨﺪ ﻣﺎﺗﺮﻳﺴﯽ‪ ،‬ﺍﺑﺮﮔﺮﺍﻓﻴﮑﯽ‪ ،‬ﺍﺯﺩﺣﺎﻡ‪ ،‬ﺗﺎﺛﻴﺮ ﻣﺤﻠﯽ‪ ،٢٨‬ﻃﺮﺍﺣﯽ ﺷﺒﮑﻪ ﻭ ﺑﺎﺯﯼﻫﺎﯼ‬
‫ﺯﻣﺎﻥﺑﻨﺪﯼ ]‪.[۱‬‬
‫‪ ٧‬ﻧﺘﻴﺠﻪ ﮔﻴﺮﯼ‬
‫ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ‪ ،‬ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﻣﻮﺭﺩ ﺑﺮﺭﺳﯽ ﻗﺮﺍﺭ ﮔﺮﻓﺖ ﻭ ﻣﺸﺨﺺ ﺷﺪ ﮐﻪ ﺍﻳﻦ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻳﮏ ﻣﻔﻬﻮﻡ ﺍﺳﺘﺎﻧﺪﺍﺭﺩ‬
‫ﺑﺮﺍﯼ ﻧﺸﺎﻥ ﺩﺍﺩﻥ ﻋﻘﻼﻳﻲ ﺑﻮﺩﻥ ﻳﮏ ﺑﺎﺯﯼ ﺍﺳﺖ ﻭ ﻧﺸﺎﻥ ﺩﻫﻨﺪﻩ ﻳﮏ ﺗﻮﺯﻳﻊ ﻋﻤﻮﻣﯽ ﺩﺭ ﻧﻤﺎﻳﻪﻫﺎﯼ ﺍﺳﺘﺮﺍﺗﮋﯼ ﺍﺳﺖ‪ .‬ﺩﺭ‬
‫ﺍﺩﺍﻣﻪ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ ﭘﻴﺪﺍ ﻧﻤﻮﺩﻥ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺩﺭ ﻣﻮﺭﺩ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﻧﻈﻴﺮ ﺑﺎﺯﯼﻫﺎﯼ ﮔﺮﺍﻓﻴﮑﯽ‪ ،‬ﭼﻨﺪ‬
‫ﻣﺎﺗﺮﻳﺴﯽ ﻣﻮﺭﺩ ﺑﺮﺭﺳﯽ ﻗﺮﺍﺭ ﮔﺮﻓﺖ ﻭ ﺍﺛﺒﺎﺕ ﺷﺪ ﮐﻪ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ ﺩﺭ ﺍﻳﻦ ﺑﺎﺯﯼﻫﺎ ﺑﻪ ﺻﻮﺭﺕ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ ﺍﺳﺖ‪.‬‬
‫ﺳﭙﺲ ﻣﺴﺎﻟﻪ ﻳﺎﻓﺘﻦ ﻧﻘﻄﻪ ﺗﻌﺎﺩﻝ ﻫﻤﺒﺴﺘﻪ ﺑﻬﻴﻨﻪ ﺑﺮﺍﯼ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ ﻣﻮﺭﺩ ﺑﺮﺭﺳﯽ ﻗﺮﺍﺭ ﮔﺮﻓﺖ ﻭ ﭼﻨﺪ ﻗﻀﻴﻪ ﺑﻴﺎﻥ ﻭ‬
‫ﻣﺸﺨﺺ ﺷﺪ‪ ،‬ﺑﻪ ﺟﺰ ﺩﺭ ﺑﺎﺯﯼﻫﺎﯼ ﮔﺮﺍﻓﻴﮑﯽ ﺑﺎ ﻋﺮﺽ ﺩﺭﺧﺘﯽ ﻣﺤﺪﻭﺩ ﻭ ﺑﺎﺯﯼﻫﺎﯼ ﺑﯽﻧﺎﻡ ﮐﻪ ﭘﻴﭽﻴﺪﮔﯽ ﭼﻨﺪ ﺟﻤﻠﻪﺍﯼ‬
‫ﺩﺍﺭﻧﺪ‪ ،‬ﺩﺭ ﺳﺎﻳﺮ ﺑﺎﺯﯼﻫﺎﯼ ﻓﺸﺮﺩﻩ‪ ،‬ﺍﻳﻦ ﻣﺴﺎﻟﻪ ﺩﺍﺭﺍﯼ ﭘﻴﭽﻴﺪﮔﯽ ﺯﻣﺎﻧﯽ ‪ NP-hard‬ﺍﺳﺖ‪.‬‬
‫ﻣﺮﺍﺟﻊ‬
‫‪1. C. H. Papadimitriou, T. Roughgarden: Computing Correlated Equilibria in Multi-player‬‬
‫)‪Games. Journal of ACM, vol. 55(3), (2008‬‬
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‫ﺯﻳﺮﻧﻮﻳﺲﻫﺎ‬
1
Correlated Equilibria
Multi-Player Games
3
Strategy Profiles
4
Polynomial-Time
5
Succinct
6
Polymatrix Games
7
Anonymous Games
8
Open Problems
9
Mixed Nash Equilibrium
10
Linear Programming
11
Linear Programming Duality
12
Ellipsoid Algorithm
13
Markov Chain Steady State
14
Signal
15
Utility
16
Chicken Game
17
Linear Maximization
18
Sparse Games
19
Type
20
Hyper-Graphical Games
21
Congestion Games
22
Polynomial Mixtures of Products
23
Linearity of Expectation
24
Polynomial Expectation Property
25
Equivalet Classes
26
Bounded Tree-Width
27
Bipartite
28
Local Effect Games
2