‫ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪﻫﺎ‬
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‫ﻣﺤﻤﺪ ﺍﻣﻴﻦ ﺑﺪﻳﻊﺯﺍﺩﮔﺎﻥ‬
‫‪[email protected]‬‬
‫ﭼﮑﻴﺪﻩ‬
‫ﻳﮑﻲ ﺍﺯ ﻣﺴﺎﺋﻠﻲ ﮐﻪ ﺩﺭ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ‪ ٢‬ﻧﻘﺶ ﺩﺍﺭﺩ‪ ،‬ﻧﺤﻮﻩ ﺗﻮﺯﻳﻊ ﺳﻮﺩ ﻳﺎ ﻫﺰﻳﻨﻪ ﺑﺎﺯﻱ ﺑﻴﻦ ﺍﻓﺮﺍﺩﻱ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﺁﻥ ﺷﺮﮐﺖ‬
‫ﻣﻲﮐﻨﻨﺪ‪ .‬ﺍﻳﻦ ﺗﻮﺯﻳﻊ ﺑﺎﻳﺪ ﺑﻪ ﮔﻮﻧﻪﺍﻱ ﺑﺎﺷﺪ ﮐﻪ ﻋﻼﻭﻩ ﺑﺮ ﺁﻧﮑﻪ ﺧﻮﺍﺳﺘﻪﻫﺎﻱ ﻫﺮ ﻓﺮﺩ ﺭﺍ ﺗﺎ ﺣﺪ ﻣﻤﮑﻦ ﺑﺮﺁﻭﺭﺩﻩ ﻣﻲﮐﻨﺪ‪ ،‬ﺧﻮﺍﺳﺘﻪﻫﺎﻱ‬
‫ﺟﻤﻌﻲ ﺭﺍ ﻧﻴﺰ ﺩﺭﺑﺮﺑﮕﻴﺮﺩ ﻭ ﻫﻤﭽﻨﻴﻦ ﻋﺎﺩﻻﻧﻪ ﺑﺎﺷﺪ‪.‬‬
‫ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ ﻣﻔﻬﻮﻡ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪﻫﺎ ﻣﻌﺮﻓﻲ ﻣﻲﺷﻮﺩ ﻭ ﻧﻘﺶ ﺁﻥ ﺩﺭ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ ﺑﻴﺎﻥ ﻣﻲﮔﺮﺩﺩ‪ .‬ﻫﻤﭽﻨﻴﻦ ﻭﻳﮋﮔﻲﻫﺎﻱ‬
‫ﺭﻭﺵﻫﺎﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺑﺮﺍﻱ ﺑﺮﺁﻭﺭﺩﻩ ﮐﺮﺩﻥ ﺧﻮﺍﺳﺘﻪﻫﺎﻱ ﺍﻓﺮﺍﺩ ﺍﺯ ﺑﺎﺯﻱ ﻭ ﺍﻋﻤﺎﻝ ﻋﺪﺍﻟﺖ ﺩﺭ ﺑﺎﺯﻱ ﺑﻴﺎﻥ ﻣﻲﺷﻮﻧﺪ‪.‬‬
‫ﻫﺰﻳﻨﻪﻫﺎ ﺩﺭ ﺑﺎﺯﻱﻫﺎ ﺑﻪ ﺩﻭ ﺻﻮﺭﺕ ﺑﺮﺭﺳﻲ ﻣﻲﺷﻮﻧﺪ‪ :‬ﻫﺰﻳﻨﻪﻫﺎﻱ ﻗﺎﺑﻞ ﺍﻧﺘﻘﺎﻝ ﻭ ﻏﻴﺮ ﻗﺎﺑﻞ ﺍﻧﺘﻘﺎﻝ‪ .‬ﺩﺭ ﺑﺎﺯﻱﻫﺎﻳﻲ ﮐﻪ ﻫﺰﻳﻨﻪ ﻳﮏ ﻓﺮﺩ‬
‫ﻗﺎﺑﻞ ﺍﻧﺘﻘﺎﻝ‪ ٣‬ﺍﺳﺖ‪ ،‬ﻫﺰﻳﻨﻪ ﺑﺎﺯﻱ ﺭﺍ ﻣﻲﺗﻮﺍﻥ ﺑﺪﻭﻥ ﻣﺤﺪﻭﺩﻳﺖ ﺑﻴﻦ ﺍﻓﺮﺍﺩ ﺑﺎﺯﻱ ﺗﻮﺯﻳﻊ ﮐﺮﺩ ﺍﻣﺎ ﺩﺭ ﺑﺎﺯﻱﻫﺎﻳﻲ ﮐﻪ ﻫﺰﻳﻨﻪﻫﺎ ﻗﺎﺑﻞ ﺍﻧﺘﻘﺎﻝ‬
‫ﻧﻴﺴﺘﻨﺪ‪ ،٤‬ﻣﺤﺪﻭﺩﻳﺖ ﺑﻴﺸﺘﺮﻱ ﺑﺮﺍﻱ ﻫﺰﻳﻨﻪ ﻳﮏ ﻓﺮﺩ ﻭﺟﻮﺩ ﺩﺍﺭﺩ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ ﺑﻴﺸﺘﺮ ﺑﻪ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ ﺑﺎ ﻗﺎﺑﻠﻴﺖ ﺍﻧﺘﻘﺎﻝ ﻫﺰﻳﻨﻪ‬
‫ﭘﺮﺩﺍﺧﺘﻪ ﻣﻲﺷﻮﺩ‪.‬‬
‫ﮐﻠﻤﺎﺕ ﮐﻠﻴﺪﻱ‪ :‬ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ‪ ،‬ﻫﺴﺘﻪ‪ ،‬ﻳﮑﻨﻮﺍﻳﻲ ﮔﺮﻭﻫﻲ‪ ،‬ﻣﮑﺎﻧﻴﺰﻡ ﮔﺮﻭﻫﻲ ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ‪ ،‬ﻣﻘﺪﺍﺭ ‪ ،Shapley‬ﻗﻀﻴﻪ ﭼﺎﻧﻪﺯﻧﻲ‬
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‫‪Cost Sharing‬‬
‫‪Cooperative Games‬‬
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‫‪Cooperative Games with Transferable Utilities‬‬
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‫‪Cooperative Games with Nontransferable Utilities‬‬
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‫‪ -۱‬ﻣﻘﺪﻣﻪ‬
‫ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ‪ ،‬ﺑﺎﺯﻱﻫﺎﻳﻲ ﻫﺴﺘﻨﺪ ﮐﻪ ﺩﺭ ﺁﻧﻬﺎ ﺭﻗﺎﺑﺖ ﺑﻪ‬
‫ﺟﺎﻱ ﺁﻧﮑﻪ ﺑﻴﻦ ﺗﮏﺗﮏ ﺍﻓﺮﺍﺩ ﺑﺎﺷﺪ ﺑﻴﻦ ﮔﺮﻭﻩﻫﺎﻳﻲ ﺍﺯ‬
‫ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﺭﻗﺎﺑﺖ ﻫﺮ ﮔﺮﻭﻩ ﺳﻮﺩﻱ ﺑﺪﺳﺖ‬
‫ﻣﻲﺁﻭﺭﺩ ﻭ ﻳﺎ ﻫﺰﻳﻨﻪﺍﻱ ﺭﺍ ﻣﺘﺤﻤﻞ ﻣﻲﺷﻮﺩ‪ .‬ﺳﻮﺍﻝ ﻣﻬﻢ ﺩﺭ ﺍﻳﻦ‬
‫ﺯﻣﻴﻨﻪ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﺍﻳﻦ ﺳﻮﺩ ﻳﺎ ﻫﺰﻳﻨﻪ ﭼﮕﻮﻧﻪ ﺑﺎﻳﺪ ﺑﻴﻦ‬
‫ﺍﻋﻀﺎﻱ ﮔﺮﻭﻩ ﺗﻮﺯﻳﻊ ﺷﻮﺩ‪ .‬ﻧﺤﻮﻩ ﭘﺎﺳﺦ ﺑﻪ ﺍﻳﻦ ﭘﺮﺳﺶ‬
‫ﻣﻮﺿﻮﻉ ﺑﺤﺚ ﺩﺭ ﺗﺌﻮﺭﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪﻫﺎ ﺩﺭ ﻧﻈﺮﻳﻪ‬
‫ﺑﺎﺯﻱﻫﺎﺳﺖ‪.‬‬
‫ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺑﻴﻦ ﺍﻋﻀﺎﻱ ﮔﺮﻭﻩ ﺑﺎﻳﺪ ﺑﻪ ﮔﻮﻧﻪﺍﻱ ﺑﺎﺷﺪ ﮐﻪ‬
‫ﺍﻫﺪﺍﻑ ﺷﺨﺼﻲ ﻭ ﮔﺮﻭﻫﻲ ﺭﺍ ﭘﻮﺷﺶ ﺩﻫﺪ ﻭ ﺩﺭ ﻋﻴﻦ ﺣﺎﻝ‬
‫ﻋﺎﺩﻻﻧﻪ ﺑﺎﺷﺪ‪ .‬ﺧﻮﺍﻫﻴﻢ ﺩﻳﺪ ﮐﻪ ﺭﻭﺵﻫﺎﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪﺍﻱ ﮐﻪ‬
‫ﺧﺼﻮﺻﻴﺖ ﻳﮑﻨﻮﺍﻳﻲ ﺑﻴﻦﮔﺮﻭﻫﻲ ﺭﺍ ﭘﺸﺘﻴﺒﺎﻧﻲ ﻣﻲﮐﻨﻨﺪ‪ ،‬ﻋﻼﻭﻩ‬
‫ﺑﺮ ﻭﻳﮋﮔﻲﻫﺎﻳﻲ ﮐﻪ ﺩﺭ ﺑﺎﻻ ﺫﮐﺮ ﺷﺪ ﻣﻲﺗﻮﺍﻧﻨﺪ ﻣﺎﻧﻊ ﺗﺒﺎﻧﻲ‬
‫ﺑﺎﺯﻳﮑﻨﺎﻥ ﺷﻮﻧﺪ‪ .‬ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻳﻦ ﺧﺼﻮﺻﻴﺖ ﻣﻲﺗﻮﺍﻥ‬
‫ﺭﻭﻳﻪﻫﺎﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪﺍﻱ ﻃﺮﺍﺣﻲ ﮐﺮﺩ ﮐﻪ ﺩﺭ ﺑﺎﺯﻱﻫﺎﻱ‬
‫ﺑﻬﻴﻨﻪﺳﺎﺯﻱ ﺗﺮﮐﻴﺐ ﻣﻮﺭﺩ ﺍﺳﺘﻔﺎﺩﻩ ﻗﺮﺍﺭ ﻣﻲﮔﻴﺮﻧﺪ‪.‬‬
‫ﺑﺮﺍﻱ ﻭﺭﻭﺩ ﺑﻪ ﺑﺤﺚ‪ ،‬ﺍﺑﺘﺪﺍ ﺑﻪ ﺗﻌﺮﻳﻒ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ ﻭ‬
‫ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪﻫﺎ ﺩﺭ ﺁﻧﻬﺎ ﻣﻲﭘﺮﺩﺍﺯﻳﻢ‪.‬‬
‫|‬
‫|‪( ) : ℙ → ℝ‬‬
‫ﺑﻪ ﻃﻮﺭﻱ ﮐﻪ ﻫﺮ ﺍﺭﺯﺵﮔﺰﺍﺭﻱ ﺑﻪ ﻣﺠﻤﻮﻋﻪﺍﻱ ﻣﺎﻧﻨﺪ ‪ S‬ﺍﺯ‬
‫ﻋﺎﻣﻞﻫﺎ‪ ،‬ﺑﺮﺩﺍﺭﻱ ﺍﺳﺖ ﺩﺭ ﻓﻀﺎﻱ | |‪ ℝ‬ﮐﻪ ﻣﻮﻟﻔﻪ ‪i‬ﺍﻡ ﺁﻥ ﺳﻮﺩ‬
‫)ﻳﺎ ﻫﺰﻳﻨﻪ( ﺑﺎﺯﻳﮑﻦ‬
‫∈ ﺭﺍ ﻣﺸﺨﺺ ﻣﻲﮐﻨﺪ‪ .‬ﺑﻪ ﻣﺠﻤﻮﻋﻪ‬
‫ﻋﺎﻣﻞﻫﺎﻱ ‪ A‬ﻭ ﺍﺭﺯﺵﮔﺰﺍﺭﻱ ‪ ،V‬ﺑﺎﺯﻱ ﺗﻌﺎﻣﻠﻲ ﻣﻲﮔﻮﻳﻴﻢ ﻭ‬
‫ﺁﻥ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ )‪ (A, V‬ﻧﺸﺎﻥ ﻣﻲﺩﻫﻴﻢ‪ .‬ﺩﺭ ﺻﻮﺭﺗﻲ ﮐﻪ‬
‫ﺍﺭﺯﺵﮔﺰﺍﺭﻱ ﻣﻮﺭﺩ ﻧﻈﺮ ﺩﺭ ﻣﻮﺭﺩ ﻫﺰﻳﻨﻪﻫﺎ ﺑﺎﺷﺪ‪ ،‬ﺁﻥ ﺭﺍ ﺑﺎ ‪c‬‬
‫ﻧﺸﺎﻥ ﻣﻲﺩﻫﻴﻢ‪ .‬ﺍﻳﻦ ﻧﻮﻉ ﺍﺭﺯﺵﮔﺰﺍﺭﻱ ﻣﺨﺼﻮﺹ ﺑﺎﺯﻱﻫﺎﻱ‬
‫ﺗﻌﺎﻣﻠﻲ ﺑﺎ ﺧﺎﺻﻴﺖ ﻏﻴﺮ ﻗﺎﺑﻞ ﺍﻧﺘﻘﺎﻝ ﺑﻮﺩﻥ‪ ٥‬ﻫﺰﻳﻨﻪﻫﺎﺳﺖ‪ .‬ﺩﺭ‬
‫ﺍﻳﻦ ﻧﻮﻉ ﺑﺎﺯﻱﻫﺎ‪ ،‬ﺳﻮﺩ ﻋﺎﻣﻞﻫﺎ ﺑﻪ ﺻﻮﺭﺕ ﺟﺪﺍﮔﺎﻧﻪ ﺩﺭ ﻧﻈﺮ‬
‫ﮔﺮﻓﺘﻪ ﻣﻲﺷﻮﺩ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﺁﻧﻬﺎ ﺭﺍ ﺑﺎ ﺑﺮﺩﺍﺭ ﻧﺸﺎﻥ ﻣﻲﺩﻫﻴﻢ‪ .‬ﺩﺭ‬
‫ﻧﻮﻉ ﺩﻳﮕﺮ ﮐﻪ ﺳﻮﺩﻫﺎ ﻗﺎﺑﻞ ﺍﻧﺘﻘﺎﻝ‪ ٦‬ﻫﺴﺘﻨﺪ‪ ،‬ﭼﻮﻥ ﻣﻲﺗﻮﺍﻥ ﺳﻮﺩ‬
‫ﮐﻠﻲ ﺭﺍ ﺑﻪ ﻫﺮ ﻧﺤﻮﻱ ﺑﻴﻦ ﺍﻋﻀﺎﻱ ﮔﺮﻭﻩ ﺗﻮﺯﻳﻊ ﮐﺮﺩ‪ ،‬ﺳﻮﺩ ﻳﮏ‬
‫ﻣﺠﻤﻮﻋﻪ ﺭﺍ ﻣﻲﺗﻮﺍﻥ ﺑﺎ ﻳﮏ ﻋﺪﺩ ﺑﻴﺎﻥ ﮐﺮﺩ‪.‬‬
‫‪( ): ℙ → ℝ‬‬
‫ﺩﺭ ﺍﻳﻨﺠﺎ ﺗﺎﺑﻊ ﺍﺭﺯﺵﮔﺰﺍﺭﻱ ﻳﮏ ﮔﺮﻭﻩ ﺍﺯ ﻋﺎﻣﻞﻫﺎ ﺑﻪ‬
‫ﻣﺠﻤﻮﻋﻪﺍﻱ ﺍﺯ ﺑﺮﺩﺍﺭﻫﺎ ﺩﺭ ﻓﻀﺎ ﻱ | |‪ ℝ‬ﺗﺒﺪﻳﻞ ﻣﻲﺷﻮﺩ ﮐﻪ‬
‫ﻣﺠﻤﻮﻉ ﻣﻮﻟﻔﻪﻫﺎﻳﺶ )ﺳﻮﺩ ﺗﮏﺗﮏ ﺑﺎﺯﻳﮑﻨﺎﻥ( ﺍﺯ ﺳﻮﺩ ﮔﺮﻭﻩ‬
‫ﺑﻴﺸﺘﺮ ﻧﻴﺴﺖ‪≤ ( ) .‬‬
‫∈ ∑ | | |‪∈ ℝ‬‬
‫=) ( ‪.‬‬
‫ﻧﻤﻮﻧﻪﺍﻱ ﺍﺯ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ‪ ،‬ﻃﺮﺍﺣﻲ ﺷﺒﮑﻪ ﺍﺗﺼﺎﻝ ﺑﻪ‬
‫‪ -۲‬ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺩﺭ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ‬
‫ﺍﻳﻨﺘﺮﻧﺖ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﺑﺎﺯﻱ ﻳﮏ ﺷﺮﮐﺖ ﺍﺭﺍﺋﻪ ﺧﺪﻣﺎﺕ‬
‫ﺩﺭ ﻳﮏ ﺑﺎﺯﻱ ﺗﻌﺎﻣﻠﻲ ﻳﮏ ﻣﺠﻤﻮﻋﻪ ﺍﺯ ﻋﺎﻣﻞﻫﺎ ﺑﻪ ﺍﺳﻢ ‪ A‬ﺑﺎ‬
‫ﺍﻳﻨﺘﺮﻧﺘﻲ ﻫﺰﻳﻨﻪ ﺍﻳﺠﺎﺩ ﺍﺭﺗﺒﺎﻁ ﻣﺸﺘﺮﻱﻫﺎ ﺑﺎ ﺍﻳﻨﺘﺮﻧﺖ ﺭﺍ ﺑﺎﻳﺪ ﺑﻪ‬
‫ﻳﮑﺪﻳﮕﺮ ﻫﻤﮑﺎﺭﻱ ﻣﻲﮐﻨﻨﺪ ﺗﺎ ﺑﻪ ﻫﺪﻑ ﻣﺸﺘﺮﮐﻲ ﺑﺮﺳﻨﺪ ﻭ ﻳﺎ‬
‫ﻃﻮﺭ ﻋﺎﺩﻻﻧﻪ ﺑﻴﻦ ﺁﻧﻬﺎ ﺗﻘﺴﻴﻢ ﮐﻨﺪ‪ .‬ﺍﻳﻦ ﻫﺰﻳﻨﻪ‪ ،‬ﻫﺰﻳﻨﻪﺍﻱ ﻗﺎﺑﻞ‬
‫ﻣﺤﺼﻮﻟﻲ ﺭﺍ ﺗﻮﻟﻴﺪ ﮐﻨﻨﺪ‪ .‬ﺧﺮﻭﺟﻲ ﺍﻳﻦ ﺗﻌﺎﻣﻞ ﺑﺎ ﻳﮏ‬
‫ﺍﻧﺘﻘﺎﻝ ﺍﺳﺖ‪.‬‬
‫ﺍﺭﺯﺵﮔﺰﺍﺭﻱ ﻣﺸﺨﺺ ﻣﻲﺷﻮﺩ‪ .‬ﺑﺮ ﻋﮑﺲ ﺑﺎﺯﻱﻫﺎﻱ‬
‫ﺩﺭ ﻧﻤﻮﻧﻪﺍﻱ ﺩﻳﮕﺮ ﻣﻲﺗﻮﺍﻥ ﻫﺰﻳﻨﻪ ﺭﺍ ﺗﺎﺧﻴﺮﻱ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺖ‬
‫ﻏﻴﺮﺗﻌﺎﻣﻠﻲ ﮐﻪ ﺧﺮﻭﺟﻲ ﺑﺮ ﺍﺳﺎﺱ ﺍﺳﺘﺮﺍﺗﮋﻱﻫﺎﻱ ﺑﺎﺯﻳﮑﻨﺎﻥ‬
‫ﮐﻪ ﻫﺮ ﻣﺸﺘﺮﻱ ﺩﺭ ﺷﺒﮑﻪ ﻣﺘﺤﻤﻞ ﻣﻲﺷﻮﺩ‪ .‬ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ‬
‫ﺑﺪﺳﺖ ﻣﻲﺁﻣﺪ‪ ،‬ﺧﺮﻭﺟﻲ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ ﺗﻨﻬﺎ ﺑﻪ‬
‫ﺑﺪﻳﻬﻲ ﺍﺳﺖ ﮐﻪ ﺍﻳﻦ ﻫﺰﻳﻨﻪ ﻗﺎﺑﻞ ﺍﻧﺘﻘﺎﻝ ﺑﻪ ﺳﺎﻳﺮ ﺑﺎﺯﻳﮑﻨﺎﻥ‬
‫ﻣﺠﻤﻮﻋﻪﺍﻱ ﺍﺯ ﻋﺎﻣﻞﻫﺎ ﮐﻪ ﺑﺎ ﻫﻢ ﺗﻌﺎﻣﻞ ﻣﻲﮐﻨﻨﺪ ﺑﺴﺘﮕﻲ ﺩﺍﺭﺩ‬
‫ﻧﻴﺴﺖ ﻭ ﻳﮏ ﻫﺰﻳﻨﻪ ﺷﺨﺼﻲ ﺍﺳﺖ‪.‬‬
‫ﺑﻪ ﻃﻮﺭﻱ ﮐﻪ ﺗﻨﻬﺎ ﺑﻪ ﺟﻨﺒﻪ ﮐﺎﺭ ﮔﺮﻭﻫﻲ ﺩﺭ ﺑﺎﺯﻱ ﭘﺮﺩﺍﺧﺘﻪ‬
‫ﺷﻮﺩ‪ .‬ﺑﻪ ﻋﺒﺎﺭﺕ ﺭﺳﻤﻲﺗﺮ‪ ،‬ﺍﺭﺯﺵﮔﺰﺍﺭﻱ ‪ V‬ﺑﺮﺍﺑﺮ ﺍﺳﺖ ﺑﺎ‪:‬‬
‫‪Cooperative Games with Nontransferable Utilities‬‬
‫‪Cooperative Games with Transferable Utilities‬‬
‫‪5‬‬
‫‪6‬‬
‫ﻧﻤﻮﻧﻪﺍﻱ ﺩﻳﮕﺮ ﺍﺯ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ ﺑﺎ ﻫﺰﻳﻨﻪ ﻗﺎﺑﻞ ﺍﻧﺘﻘﺎﻝ‪،‬‬
‫ﺑﺎﺯﻱ ﻣﺮﺍﮐﺰ ﺧﺪﻣﺎﺕ ﺍﺳﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﻣﺴﺎﻟﻪ‪ ،‬ﻗﺮﺍﺭ ﺍﺳﺖ ﻳﮏ‬
‫ﻣﺠﻤﻮﻋﻪ ﺍﺯ ﻣﺮﺍﮐﺰ ﺧﺪﻣﺎﺕﺩﻫﻲ )‪ (F‬ﺑﻪ ﻳﮏ ﻣﺠﻤﻮﻋﻪ ﺍﺯ‬
‫ﻣﺸﺘﺮﻱﻫﺎ )‪ (A‬ﺧﺪﻣﺘﻲ ﺍﺭﺍﺋﻪ ﺩﻫﻨﺪ‪ .‬ﻣﺸﺘﺮﻱﻫﺎ ﻭ ﻣﺮﺍﮐﺰ‬
‫ﺧﺪﻣﺎﺗﻲ ﺑﻪ ﺻﻮﺭﺕ ﮔﺮﺍﻓﻲ ﺑﻪ ﻳﮑﺪﻳﮕﺮ ﻣﺘﺼﻠﻨﺪ ﺑﻪ ﻃﻮﺭﻱ ﮐﻪ‬
‫ﻓﺎﺻﻠﻪ ﺁﻧﻬﺎ ﺍﺯ ﻳﮑﺪﻳﮕﺮ ﺑﺎ ﻣﺎﺗﺮﻳﺲ ‪ d‬ﻣﺸﺨﺺ ﺷﺪﻩ ﺍﺳﺖ‪.‬‬
‫ﻓﻌﺎﻝﮐﺮﺩﻥ ﻫﺮ ﻣﺮﮐﺰ ﮐﻪ ﺁﻥ ﺭﺍ ﺑﺎ ‪ i‬ﻧﺸﺎﻥ ﻣﻲﺩﻫﻴﻢ ﻫﺰﻳﻨﻪﺍﻱ ﺩﺭ‬
‫ﺑﺮ ﺩﺍﺭﺩ ﮐﻪ ﺑﺎ ‪ fi‬ﻣﺸﺨﺺ ﻣﻲﺷﻮﺩ‪ .‬ﻫﺪﻑ ﺩﺭ ﺍﻳﻦ ﺑﺎﺯﻱ ﺍﻳﻦ‬
‫ﺍﺳﺖ ﮐﻪ ﺗﻤﺎﻡ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺧﺪﻣﺎﺕ ﺩﺭﻳﺎﻓﺖ ﮐﻨﻨﺪ ﺑﻪ ﻃﻮﺭﻱ ﮐﻪ‬
‫ﻫﺰﻳﻨﻪ ﮐﻞ ﻣﻴﻨﻴﻤﻢ ﺷﻮﺩ‪.‬‬
‫ﺑﺮﺍﻱ ﺑﺮﺭﺳﻲ ﻧﺤﻮﻩ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺩﺭ ﻳﮏ ﺑﺎﺯﻱ‪ ،‬ﺍﺑﺘﺪﺍ ﺑﺎﻳﺪ ﺑﻪ‬
‫ﺗﻌﺮﻳﻒ ﻫﺴﺘﻪ ﺑﺎﺯﻱ ﭘﺮﺩﺍﺧﺖ‪.‬‬
‫ﻫﺴﺘﻪ ﻳﮏ ﺑﺎﺯﻱ‪ ،‬ﺧﺮﻭﺟﻲﺍﻱ ﺍﺯ ﺑﺎﺯﻱ ﺗﻤﺎﻡ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺍﺳﺖ ﺑﻪ‬
‫ﻃﻮﺭﻱ ﮐﻪ ﻫﻴﭻ ﮔﺮﻭﻫﻲ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻧﻔﻊ ﺑﻴﺸﺘﺮﻱ ﺍﺯ ﺷﮑﺴﺘﻦ‬
‫ﺍﺋﺘﻼﻑ ﮐ ﻞ‬
‫ﻭ ﺗﺸﮑﻴﻞ ﮔﺮﻭﻫﻲ ﺟﺪﺍﮔﺎﻧﻪ ﻧﻤﻲﺑﺮﻧﺪ‪ .‬ﺑﻪ‬
‫ﺻﻮﺭﺕ ﺭﺳﻤﻲﺗﺮ ﺍﮔﺮ )‪ (A, c‬ﻳﮏ ﺑﺎﺯﻱ ﺗﻌﺎﻣﻠﻲ ﺑﺎ ﻫﺰﻳﻨﻪ ﻗﺎﺑﻞ‬
‫ﺍﻧﺘﻘﺎﻝ ﺑﺎﺷﺪ‪ ،‬ﺑﻪ‬
‫|‬
‫|‪∈ ℝ‬‬
‫ﻫﺴﺘﻪ ﺑﺎﺯﻱ ﻣﻲﮔﻮﻳﻨﺪ ﺍﮔﺮ ﺍﻳﻦ‬
‫ﺑﺮﺩﺍﺭ ﺧﻮﺍﺹ ﺯﻳﺮ ﺭﺍ ﺩﺍﺭﺍ ﺑﺎﺷﺪ‪:‬‬
‫·‬
‫ﮐﻪ ﻫﺰﻳﻨﻪ‬
‫ﺍﺧﺘﺼﺎﺹﻳﺎﻓﺘﻪ ﺑﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺍﺳﺖ ﺑﺎﻳﺪ ﺑﺎ ﻫﺰﻳﻨﻪ ﮐﻞ‬
‫·‬
‫ﻫﺰﻳﻨﻪ ﺭﺍ ﻃﻮﺭﻱ ﺑﻴﻦ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺗﻘﺴﻴﻢ ﻣﻲﮐﻨﺪ ﮐﻪ ﺑﻪ ﻧﻔﻊ ﻫﻤﻪ‬
‫ﺑﺎﺷﺪ‪ .‬ﺍﻣﺎ ﻣﻮﺍﻗﻌﻲ ﻧﻴﺰ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﮐﻪ ﻫﺴﺘﻪ ﺑﺎﺯﻱ ﺗﻬﻲ ﺍﺳﺖ‪.‬‬
‫ﺑﺮﺍﻱ ﺑﺮﺭﺳﻲ ﺗﻬﻲ ﺑﻮﺩﻥ ﻫﺴﺘﻪ ﻳﮏ ﺑﺎﺯﻱ‪Bondareva- ،‬‬
‫‪ Shapley‬ﺷﺮﻁ ﻻﺯﻡ ﻭ ﮐﺎﻓﻲﺍﻱ ﺭﺍ ﺍﺭﺍﺋﻪ ﮐﺮﺩﻩﺍﻧﺪ‪ .‬ﺑﺮﺍﻱ ﺑﻴﺎﻥ‬
‫ﺍﻳﻦ ﺷﺮﻁ ﺍﺑﺘﺪﺍ ﻣﺠﻤﻮﻋﻪ ﻭﺯﻥﻫﺎﻱ ﻣﺘﻮﺍﺯﻥ ﺭﺍ ﺗﻌﺮﻳﻒ ﻣﻲﮐﻨﻴﻢ‪.‬‬
‫ﺗﻌﺮﻳﻒ‪ :۱‬ﺑﺮﺩﺍﺭ‬
‫ﮐﻪ ﺑﻪ ﻫﺮ ﺯﻳﺮﻣﺠﻤﻮﻋﻪ ﺍﺯ ‪ A‬ﻳﮏ ﻭﺯﻥ‬
‫ﻧﺴﺒﺖ ﻣﻲﺩﻫﺪ ﺭﺍ ﻳﮏ ﻣﺠﻤﻮﻋﻪ ﻭﺯﻥ ﻣﺘﻮﺍﺯﻥ ﻣﻲﻧﺎﻣﻴﻢ ﺍﮔﺮ ﺑﻪ‬
‫ﺍﺯﺍﻱ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ‪ j‬ﻭ ﻫﺮ‬
‫ﮐﻪ ‪ j‬ﺭﺍﺷﺎﻣﻞ ﺑﺎﺷﺪ‪ ،‬ﻣﺠﻤﻮﻉ‬
‫⊆‬
‫ﻭﺯﻥﻫﺎﻱ ﻧﺴﺒﺖ ﺩﺍﺩﻩ ﺷﺪﻩ ﺑﻪ ﻣﺠﻤﻮﻋﻪ ﻫﺎﻱ ‪ S‬ﺑﺮﺍﺑﺮ ‪ ۱‬ﺑﺎﺷﺪ‪.‬‬
‫∈ ‪⊆ :‬‬
‫∙‬
‫∈ ∀‬
‫ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻳﻦ ﺗﻌﺮﻳﻒ‪ ،‬ﺷﺮﻁ ﻣﺮﺑﻮﻁ ﺑﻪ ﺗﻬﻲ ﻧﺒﻮﺩﻥ ﻫﺴﺘﻪ‬
‫ﺭﺍ ﺩﺭ ﻗﻀﻴﻪ ﺯﻳﺮ ﺑﻴﺎﻥ ﻣﻲﮐﻨﻴﻢ‪.‬‬
‫ﻗﻀﻴﻪ‪ :۱‬ﺩﺭ ﻳﮏ ﺑﺎﺯﻱ ﺗﻌﺎﻣﻠﻲ ﺑﺎ ﻗﺎﺑﻠﻴﺖ ﺍﻧﺘﻘﺎﻝ ﺳﻮﺩ‪ ،‬ﻫﺴﺘﻪ‬
‫ﺑﺎﺯﻱ ﺗﻬﻲ ﻧﻴﺴﺖ ﺍﮔﺮ ﻭ ﻓﻘﻂ ﺍﮔﺮ ﺑﺮﺍﻱ ﻫﺮ ﻣﺠﻤﻮﻋﻪ ﻭﺯﻥﻫﺎﻱ‬
‫ﻣﺘﻮﺍﺯﻥ ﻣﺜﻞ‬
‫ﺩﺍﺷﺘﻪ ﺑﺎﺷﻴﻢ‪( ) ≥ ( ) :‬‬
‫⊆‬
‫∑‬
‫ﺍﺛﺒﺎﺕ‪ :‬ﻣﻲﺗﻮﺍﻥ ﺩﻭ ﻭﻳﮋﮔﻲ ﻫﺴﺘﻪ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﻳﮏ ﻣﺴﺎﻟﻪ‬
‫‪٨‬‬
‫ﺗﻮﺍﺯﻥ ﭘﻮﻟﻲ‪ :‬ﻣﺠﻤﻮﻉ ﻣﻮﻟﻔﻪﻫﺎﻱ ﺑﺮﺩﺍﺭ‬
‫ﺑﺎﺯﻱ ﺑﺮﺍﺑﺮ ﺑﺎﺷﺪ‪= ( ).‬‬
‫∈‬
‫∙‬
‫ﻫﻤﺎﻥﻃﻮﺭ ﮐﻪ ﺩﻳﺪﻳﻢ‪ ،‬ﻫﺴﺘﻪ ﻳﮏ ﺑﺎﺯﻱ ﺧﺮﻭﺟﻲﺍﻱ ﺍﺳﺖ ﮐﻪ‬
‫‪=1‬‬
‫‪ -۳‬ﻫﺴﺘﻪ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ‬
‫‪٧‬‬
‫) ( ≤‬
‫⊆ ∀‬
‫∈ ∑‪.‬‬
‫ﺑﺮﻧﺎﻣﻪﻧﻮﻳﺴﻲ ﺧﻄﻲ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺖ‪ .‬ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ ﺍﮔﺮ ﺍﻳﻦ‬
‫‪ LP‬ﺟﻮﺍﺑﻲ ﺑﺮﺍﺑﺮ ﺑﺎ )‪ c(A‬ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ ﺁﻧﮕﺎﻩ ﻫﺴﺘﻪ ﺑﺎﺯﻱ ﺗﻬﻲ‬
‫ﻧﺨﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬
‫ﻭﻳﮋﮔﻲ ﻫﺴﺘﻪ‪ :‬ﺑﺮﺍﻱ ﻫﺮ ﺯﻳﺮﻣﺠﻤﻮﻋﻪﺍﻱ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ‪،‬‬
‫ﻣﺠﻤﻮﻉ ﻫﺰﻳﻨﻪ ﺍﺧﺘﺼﺎﺹﻳﺎﻓﺘﻪ ﺑﻪ ﺗﮏﺗﮏ ﺁﻧﻬﺎ ﻧﺒﺎﻳﺪ‬
‫ﺑﻴﺸﺘﺮ ﺍﺯ ﻫﺰﻳﻨﻪﺍﻱ ﺑﺎﺷﺪ ﮐﻪ ﺑﻪ ﮐﻞ ﮔﺮﻭﻩ ﺗﻌﻠﻖ ﻣﻲﮔﻴﺮﺩ‪.‬‬
‫ﺑﺎ ﺍﻳﻦ ﻭﻳﮋﮔﻲ ﻫﻴﭻ ﮐﺲ ﻧﻤﻲﺧﻮﺍﻫﺪ ﮔﺮﻭﻩ ﺑﺰﺭﮒ ﺭﺍ‬
‫ﺗﺮﮎ ﮐﻨﺪ‪.‬‬
‫‪Grand Coalition‬‬
‫‪7‬‬
‫‪≤ ( ).‬‬
‫∈‬
‫∙‬
‫∈‬
‫⊆ ∀‬
‫ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻗﻀﻴﻪ ‪ LP‬ﻫﻤﺰﺍﺩ ﻗﻮﻱ‪ ،‬ﺍﮔﺮ ‪ LP‬ﻗﻮﻱ ﻫﻤﺰﺍﺩ ﺍﻳﻦ‬
‫ﻣﺴﺎﻟﻪ ﺭﺍ ﺑﻨﻮﻳﺴﻴﻢ‪ ،‬ﺟﻮﺍﺏ ﺁﻥ ﻧﻴﺰ ﺑﺎﻳﺪ ﺑﺮﺍﺑﺮ ﺑﺎ )‪ c(A‬ﺑﺎﺷﺪ‪.‬‬
‫)‪Linear Programming (LP‬‬
‫‪8‬‬
‫‪ -۴‬ﻣﮑﺎﻧﻴﺰﻡﻫﺎﻱ ﮔﺮﻭﻫﻲ ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﻭ ﺭﻭﻳﻪﻫﺎﻱ ﺗﻮﺯﻳﻊ‬
‫) (‬
‫‪=1‬‬
‫∈ ‪⊆ :‬‬
‫‪≥ 0.‬‬
‫⊆‬
‫∙‬
‫ﻫﺰﻳﻨﻪ ﺑﻴﻦﮔﺮﻭﻫﻲ ﻳﮑﻨﻮﺍ‬
‫∈ ∀‬
‫∙‬
‫ﺩﺭ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ ﮐﻪ ﺗﻘﺎﺿﺎ ﻧﺴﺒﺖ ﺑﻪ ﻗﻴﻤﺖ ﺣﺴﺎﺱ‬
‫⊆ ∀‬
‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻫﺴﺘﻪ ﺑﺎﺯﻱ ﺗﻬﻲ ﻧﺨﻮﺍﻫﺪ ﺑﻮﺩ ﺍﮔﺮ ﻭ ﻓﻘﻂ ﺍﮔﺮ ﺑﺮﺍﻱ‬
‫ﻫﺮ ﻣﺠﻤﻮﻋﻪ ﻭﺯﻥﻫﺎﻱ ﻣﺘﻮﺍﺯﻥ ﻣﺜﻞ‬
‫) ( ≥) (‬
‫⊆‬
‫ﻫﺴﺘﻪ ﺗﻘﺮﻳﺒﻲ‬
‫ﺩﺍﺷﺘﻪ ﺑﺎﺷﻴﻢ‪:‬‬
‫∑‬
‫ﺑﺎﺯﻱﻫﺎﻱ ﺑﻬﻴﻨﻪﺳﺎﺯﻱ ﺗﺮﮐﻴﺐ ﮐﻪ ﻣﻌﻤﻮﻻ ﭘﻴﭽﻴﺪﮔﻲ ﻣﺤﺎﺳﺒﺎﺗﻲ‬
‫ﺩﺍﺭﻧﺪ‪ ،‬ﺭﺥ ﻣﻲﺩﻫﺪ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﻫﺴﺘﻪ ﺁﻧﻬﺎ ﺗﻬﻲ ﺍﺳﺖ‪ .‬ﺑﻪ‬
‫ﻋﻼﻭﻩ ﺩﺭ ﺑﺮﺧﻲ ﻣﻮﺍﺭﺩ‪ ،‬ﭘﻴﺪﺍ ﮐﺮﺩﻥ ﺍﻳﻨﮑﻪ ﻫﺴﺘﻪ ﺗﻬﻲ ﺍﺳﺖ ﻳﺎ‬
‫ﺧﻴﺮ ﺑﻪ ﺧﺎﻃﺮ ﭘﻴﭽﻴﺪﮔﻲ ﻣﺤﺎﺳﺒﺎﺗﻲ ﻋﻤﻼ ﻏﻴﺮﻣﻤﮑﻦ ﻣﻲﺷﻮﺩ‪.‬‬
‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺮﺍﻱ ﭘﻴﺪﺍ ﮐﺮﺩﻥ ﻳﮏ ﺗﻮﺯﻳﻊ ﻓﺮﺍﮔﻴﺮ ﺗﻘﺮﻳﺒﻲ ﻭ ﻧﻪ‬
‫ﺗﺤﻘﻴﻘﻲ ﺗﻌﺮﻳﻔﻲ ﺩﻳﮕﺮ ﺍﺯ ﻫﺴﺘﻪ‪ ،‬ﺑﺎ ﻧﺎﻡ ﻫﺴﺘﻪ ﺗﻘﺮﻳﺒﻲ ﺍﺭﺍﺋﻪ‬
‫ﻣﻲﮐﻨﻴﻢ‪.‬‬
‫ﺗﻌﺮﻳﻒ‪ :۲‬ﻫﺴﺘﻪ ﺗﻘﺮﻳﺒﻲ ﺑﺎ ﺿﺮﻳﺐ‬
‫)‪ (A, c‬ﺑﺮﺩﺍﺭ‬
‫ﺑﺮﺁﻭﺭﺩﻩ ﮐﻨﺪ‪.‬‬
‫ﺧﺪﻣﺎﺕ ﻣﻮﺭﺩ ﻧﻈﺮ ﺭﺍ ﺑﻪ ﻣﺰﺍﻳﺪﻩ ﺑﮕﺬﺍﺭﺩ ﺗﺎ ﺑﺪﻳﻦ ﻭﺳﻴﻠﻪ‬
‫ﻣﺸﺘﺮﻱﻫﺎﻳﻲ ﺭﺍ ﮐﻪ ﺗﻮﺍﻥ ﭘﺮﺩﺍﺧﺖ ﻫﺰﻳﻨﻪ ﺭﺍ ﺩﺍﺭﻧﺪ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ‬
‫ﺳﺎﺧﺘﺎﺭ ﻣﺎﻟﻲ ﻣﻌﺎﻣﻠﻪ ﺍﻧﺘﺨﺎﺏ ﮐﻨﺪ ﻭ ﺑﻪ ﺍﻳﻦ ﺗﺮﺗﻴﺐ ﻗﻴﻤﺖ‬
‫ﻣﺸﮑﻠﻲ ﮐﻪ ﺩﺭ ﻣﻮﺭﺩ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ‪ ،‬ﻣﺨﺼﻮﺻﺎ‬
‫|‬
‫ﺍﺳﺖ‪ ،‬ﺳﺮﻭﻳﺲﺩﻫﻨﺪﻩ ﺑﻪ ﺟﺎﻱ ﺗﻌﻴﻴﻦ ﻗﻴﻤﺖ‪ ،‬ﻣﻲﺗﻮﺍﻧﺪ ﮐﺎﻻ ﻳﺎ‬
‫ﻳﮏ ﺑﺎﺯﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ‬
‫ﺍﺳﺖ ﺍﮔﺮ ﺩﻭ ﻭﻳﮋﮔﻲ ﺯﻳﺮ ﺭﺍ‬
‫|‪∈ ℝ‬‬
‫ﻗﺎﺑﻞﻗﺒﻮﻝﺗﺮﻱ ﺍﺯ ﻧﻈﺮ ﻫﻤﻪ ﺑﺪﺳﺖ ﺁﻳﺪ‪ .‬ﺩﺭ ﺍﻳﻨﺠﺎ ﻫﺪﻑ‬
‫ﻃﺮﺍﺣﻲ ﻳﮏ ﻣﺰﺍﻳﺪﻩ ﺍﺳﺖ ﺑﻪ ﻃﻮﺭﻱ ﮐﻪ ﻋﻼﻭﻩ ﺑﺮ ﻣﻨﺎﻓﻊ‬
‫ﺷﺨﺼﻲ‪ ،‬ﻣﻨﺎﻓﻊ ﺟﻤﻌﻲ ﻧﻴﺰ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﺷﻮﺩ ﺗﺎ ﺍﻓﺮﺍﺩ‬
‫ﭘﻴﺸﻨﻬﺎﺩﺍﺕ ﻭﺍﻗﻌﻲ ﺧﻮﺩ ﺭﺍ ﺩﺭ ﻣﻌﺎﻣﻠﻪ ﻣﻄﺮﺡ ﮐﻨﻨﺪ‪ .‬ﺑﺮﺍﻱ ﺍﺩﺍﻣﻪ‬
‫ﺑﺤﺚ‪ ،‬ﺍﺑﺘﺪﺍ ﻣﺴﺎﻟﻪ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺭﺳﻤﻲ ﺑﻴﺎﻥ ﻣﻲﮐﻨﻴﻢ‪.‬‬
‫ﺩﺭ ﺍﻳﻦ ﻣﺴﺎﻟﻪ‪ A ،‬ﺭﺍ ﻣﺠﻤﻮﻋﻪﺍﻱ ﺍﺯ ‪ n‬ﺑﺎﺯﻳﮑﻨﻲ ﺩﺭ ﻧﻈﺮ‬
‫ﻣﻲﮔﻴﺮﻳﻢ ﮐﻪ ﺩﺭ ﺍﻧﺘﻈﺎﺭ ﺳﺮﻭﻳﺲ ﻫﺴﺘﻨﺪ‪ .‬ﺗﺎﺑﻊ ﺗﺨﺼﻴﺺ‬
‫ﻫﺰﻳﻨﻪ ﺑﻪ ﻫﺮ ﺯﻳﺮﻣﺠﻤﻮﻋﻪ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ‪ ،‬ﺑﻪ ﺻﻮﺭﺕ‬
‫}‪ : ℙ → ℝ ∪ {0‬ﺗﻌﺮﻳﻒ ﻣﻲﺷﻮﺩ‪ .‬ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺧﺪﻣﺎﺕ‬
‫ﺑﺮﺍﻱ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ‪ i‬ﺑﻪ ﺍﻧﺪﺍﺯﻩ ‪ ui‬ﻣﻲﺍﺭﺯﺩ‪ .‬ﺍﮔﺮ ﺩﺭ ﻧﻬﺎﻳﺖ‬
‫ﺑﺎﺯﻳﮑﻦ ‪ i‬ﺳﺮﻭﻳﺲ ﮔﺮﻓﺖ‪ qi ،‬ﺁﻥ ‪ ۱‬ﻣﻲﺷﻮﺩ ﻭ ﺩﺭ ﻏﻴﺮ ﺍﻳﻦ‬
‫ﺻﻮﺭﺕ ﻣﻘﺪﺍﺭ ﺻﻔﺮ ﻣﻲﮔﻴﺮﺩ‪ xi .‬ﻧﻴﺰ ﻣﺸﺨﺺ ﻣﻲﮐﻨﺪ ﮐﻪ‬
‫‪≤ ( ):‬‬
‫·‬
‫ﺗﻮﺍﺯﻥ ﭘﻮﻟﻲ‬
‫·‬
‫ﻭﻳﮋﮔﻲ ﻫﺴﺘﻪ‪≤ ( ) :‬‬
‫∈‬
‫∈‬
‫∑≤) (‬
‫∑∙‬
‫⊆ ∀‬
‫ﺑﺎﺯﻳﮑﻦ ﭼﻪ ﻣﻘﺪﺍﺭ ﺑﺎﻳﺪ ﻫﺰﻳﻨﻪ ﮐﻨﺪ ﺗﺎ ﺍﺯ ﺧﺪﻣﺎﺕ ﺍﺳﺘﻔﺎﺩﻩ ﮐﻨﺪ‪.‬‬
‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻣﻨﻔﻌﺖ ﺑﺎﺯﻳﮑﻦ ‪i‬ﺍﻡ ﺍﺯ ﻓﺮﻣﻮﻝ‬
‫‪−‬‬
‫ﻣﺤﺎﺳﺒﻪ‬
‫ﺑﺎ ﺍﻳﻦ ﺗﻌﺮﻳﻒ‪ ،‬ﻗﻀﻴﻪ ‪ Bondareva-Shapley‬ﺩﺭ ﻣﻮﺭﺩ‬
‫ﻣﻲﺷﻮﺩ‪.‬‬
‫ﺷﺮﻁ ﺗﻬﻲ ﻧﺒﻮﺩﻥ ﻫﺴﺘﻪ‪ ،‬ﺑﻪ ﺻﻮﺭﺕ ﮐﻠﻲﺗﺮ ﺯﻳﺮ ﺗﺒﺪﻳﻞ‬
‫ﻳﮏ ﻣﮑﺎﻧﻴﺰﻡ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ‪ ،‬ﺑﺎ ﺩﺭﻳﺎﻓﺖ ﭘﻴﺸﻨﻬﺎﺩﺍﺕ ﻫﺮ‬
‫ﻣﻲﺷﻮﺩ‪.‬‬
‫ﺑﺎﺯﻳﮑﻦ‪ ،‬ﺗﺼﻤﻴﻢ ﻣﻲﮔﻴﺮﺩ ﮐﻪ ﭼﻪ ﮐﺴﺎﻧﻲ ﺑﺎﻳﺪ ﺳﺮﻭﻳﺲ ﺑﮕﻴﺮﻧﺪ‬
‫ﻗﻀﻴﻪ‪ :۲‬ﺑﺮﺍﻱ ﻫﺮ ‪≥ 1‬‬
‫ﻳﮏ ﺑﺎﺯﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ )‪ (A, c‬ﺑﺎ‬
‫ﻗﺎﺑﻠﻴﺖ ﺍﻧﺘﻘﺎﻝ ﻫﺰﻳﻨﻪ ﺩﺍﺭﺍﻱ ﻫﺴﺘﻪ ﻧﺎﺗﻬﻲ ﺍﺳﺖ ﺍﮔﺮ ﻭ ﻓﻘﻂ ﺍﮔﺮ‬
‫ﺑﺮﺍﻱ ﻫﺮ ﻣﺠﻤﻮﻋﻪ ﻭﺯﻥﻫﺎﻱ ﻣﺘﻮﺍﺯﻥ ﻣﺜﻞ‬
‫) (‬
‫≥) (‬
‫⊆‬
‫∑‬
‫ﺩﺍﺷﺘﻪ ﺑﺎﺷﻴﻢ‪:‬‬
‫ﻭ ﭼﻪ ﻣﻘﺪﺍﺭ ﺑﺎﻳﺪ ﭘﺮﺩﺍﺧﺖ ﮐﻨﻨﺪ‪ .‬ﻣﺠﻤﻮﻋﻪ ﺑﺎﺯﻳﮑﻨﺎﻧﻲ ﮐﻪ‬
‫ﺳﺮﻭﻳﺲ ﺭﺍ ﺩﺭﻳﺎﻓﺖ ﻣﻲﮐﻨﻨﺪ ﺑﺎ ‪ Q‬ﻭ ﻫﺰﻳﻨﻪ ﭘﺮﺩﺍﺧﺘﻲ ﺁﻧﻬﺎ ﺭﺍ‬
‫ﺑﺎ ﺑﺮﺩﺍﺭ ‪ p‬ﻧﻤﺎﻳﺶ ﻣﻲﺩﻫﻴﻢ‪ .‬ﺑﺮﺍﻱ ﺍﻳﻦ ﻣﮑﺎﻧﻴﺰﻡ ﺷﺮﻁﻫﺎﻱ ﺯﻳﺮ‬
‫ﺭﺍ ﻧﻴﺰ ﺩﺭ ﻧﻈﺮ ﻣﻲﮔﻴﺮﻳﻢ‪.‬‬
‫·‬
‫ﺑﺪﻭﻥ ﺭﺷﻮﻩ‪ :‬ﻫﺰﻳﻨﻪ ﭘﺮﺩﺍﺧﺘﻲ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﻣﻨﻔﻲ ﻧﻴﺴﺖ‪.‬‬
‫)‪≥ 0‬‬
‫(‬
‫·‬
‫·‬
‫ﺷﺮﮐﺖ ﺩﺍﻭﻃﻠﺒﺎﻧﻪ‪ :‬ﺍﮔﺮ ﮐﺴﻲ ﺳﺮﻭﻳﺲ ﻧﻤﻲﮔﻴﺮﺩ‪ ،‬ﭘﻮﻟﻲ‬
‫ﺩﺭ ﺭﻭﺷﻲ ﮐﻪ ﺑﻴﻦﮔﺮﻭﻫﻲ ﻳﮑﻨﻮﺍ ﺍﺳﺖ ﺍﮔﺮ ﮔﺮﻭﻩ ﮔﺴﺘﺮﺵ‬
‫ﻧﻴﺰ ﭘﺮﺩﺍﺧﺖ ﻧﺨﻮﺍﻫﺪ ﮐﺮﺩ ﻭ ﺍﮔﺮ ﺑﻪ ﮐﺴﻲ ﺳﺮﻭﻳﺲ ﺩﺍﺩﻩ‬
‫ﭘﻴﺪﺍ ﮐﻨﺪ ﻭ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺩﻳﮕﺮﻱ ﺭﺍ ﺷﺎﻣﻞ ﺷﻮﺩ‪ ،‬ﻫﺰﻳﻨﻪﺍﻱ ﮐﻪ ﺩﺭ‬
‫ﻣﻲﺷﻮﺩ‪ ،‬ﻫﺰﻳﻨﻪ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺳﺮﻭﻳﺲ‪ ،‬ﺍﺯ ﺍﺭﺯﺷﻲ ﮐﻪ ﺁﻥ‬
‫ﺣﺎﻟﺖ ﺟﺪﻳﺪ ﺑﻪ ﻳﮏ ﻓﺮﺩ ﺗﺤﻤﻴﻞ ﻣﻲﺷﻮﺩ ﺍﺯ ﺣﺎﻟﺖ ﻗﺒﻠﻲﺍﺵ‬
‫ﺧﺪﻣﺖ ﺑﺮﺍﻱ ﺷﺨﺺ ﺩﺍﺭﺩ ﺑﻴﺸﺘﺮ ﻧﻴﺴﺖ‪.‬‬
‫ﺑﺪﺗﺮ ﻧﺨﻮﺍﻫﺪ ﺑﻮﺩ‪ .‬ﺑﻪ ﻋﺒﺎﺭﺕ ﺩﻳﮕﺮ‪ ،‬ﺩﺭ ﺍﻳﻦ ﻧﻮﻉ ﺭﻭﺵﻫﺎ‪ ،‬ﮐﺎﺭ‬
‫ﻣﺸﺘﺮﻱ ﻣﺤﻮﺭﻱ‪ :‬ﺑﺮﺍﻱ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ‪ ،i‬ﭘﻴﺸﻨﻬﺎﺩﻱ ﻣﺎﻧﻨﺪ‬
‫ﮔﺮﻭﻫﻲ ﺍﮔﺮ ﺑﺎﻋﺚ ﮐﺎﻫﺶ ﻫﺰﻳﻨﻪ ﺍﻓﺮﺍﺩ ﮔﺮﻭﻩ ﻧﺸﻮﺩ‪ ،‬ﺑﺎﻋﺚ‬
‫ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﮐﻪ ﺍﮔﺮ ﺁﻥ ﺭﺍ ﺩﺭ ﻣﺰﺍﻳﺪﻩ ﻣﻄﺮﺡ ﮐﻨﺪ‪،‬‬
‫ﺍﻓﺰﺍﻳﺶ ﺁﻥ ﻧﺨﻮﺍﻫﺪ ﺷﺪ‪ .‬ﺑﺮﺍﻱ ﺗﻌﺮﻳﻒ ﺭﺳﻤﻲ ﻳﮑﻨﻮﺍﻳﻲ‬
‫∗‬
‫·‬
‫ﺑﺪﻭﻥ ﺗﻮﺟﻪ ﺑﻪ ﭘﻴﺸﻨﻬﺎﺩﺍﺕ ﺑﻘﻴﻪ ﭘﻴﺮﻭﺯ ﻣﻲﺷﻮﺩ‪.‬‬
‫ﺑﻴﻦﮔﺮﻭﻫﻲ ﺍﺑﺘﺪﺍ ﺑﺎﻳﺪ ﺭﻭﻳﻪ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺭﺍ ﺗﻌﺮﻳﻒ ﮐﻨﻴﻢ‪.‬‬
‫ﺗﻮﺍﺯﻥ ﭘﻮﻟﻲ ﺗﻘﺮﻳﺒﻲ‪ :‬ﻣﮑﺎﻧﻴﺰﻡ ﺑﺎﻳﺪ ﺍﺯ ﻧﻈﺮ ﺑﻮﺩﺟﻪ ﺑﻪ ﻃﻮﺭ‬
‫ﺗﻌﺮﻳﻒ‪ :۳‬ﺍﮔﺮ )‪ (A, c‬ﻳﮏ ﺑﺎﺯﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺑﺎﺷﺪ‪ ،‬ﺁﻧﮕﺎﻩ‬
‫ﺗﻘﺮﻳﺒﻲ ﻣﺘﻮﺍﺯﻥ ﺑﺎﺷﺪ ﻳﻌﻨﻲ‪:‬‬
‫) ( ≤‬
‫∈‬
‫ﻳﮏ ﺭﻭﻳﻪ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﻳﮏ ﺗﺎﺑﻊ ﺗﺨﺼﻴﺺ ﻫﺰﻳﻨﻪ ﺍﺳﺖ ﮐﻪ ﺑﻪ‬
‫≤) (‬
‫ﮐﺴﺎﻧﻲ ﮐﻪ ﺩﺭ ﮔﺮﻭﻩ ﻧﻴﺴﺘﻨﺪ ﻣﻘﺪﺍﺭ ﺻﻔﺮ ﻭ ﺑﻪ ﺑﻘﻴﻪ ﻣﻘﺪﺍﺭ‬
‫ﻋﻼﻭﻩ ﺑﺮ ﻭﻳﮋﮔﻲﻫﺎﻱ ﺑﺎﻻ ﺑﺮﺍﻱ ﺍﻳﻨﮑﻪ ﻣﮑﺎﻧﻴﺰﻣﻲ ﮔﺮﻭﻫﻲ‬
‫ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﺑﺎﺷﺪ‪ ،‬ﺑﺎﻳﺪ ﺍﺯ ﻭﻳﮋﮔﻲ ﺯﻳﺮ ﻧﻴﺰ ﺑﺮﺧﻮﺭﺩﺍﺭ ﺑﺎﺷﺪ‪.‬‬
‫ﻓﺮﺽ ﮐﻨﻴﺪ ﮐﻪ‬
‫⊆‬
‫ﻳﮏ ﮔﺮﻭﻩ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺑﺎﺷﻨﺪ‪ .‬ﺩﻭ‬
‫ﺑﺮﺩﺍﺭ ﭘﻴﺸﻨﻬﺎﺩﺍﺕ ‪ u‬ﻭ ’‪ u‬ﺭﺍ ﺩﺭ ﻧﻈﺮ ﻣﻲﮔﻴﺮﻳﻢ ﮐﻪ ﺣﺎﻭﻱ‬
‫ﭘﻴﺸﻨﻬﺎﺩﺍﺕ ﺗﻤﺎﻡ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺩﺭ ﺩﻭ ﺣﺎﻟﺖ ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﻭ‬
‫ﻏﻴﺮﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﺍﺳﺖ‪ .‬ﺍﻳﻦ ﺩﻭ ﺑﺮﺩﺍﺭ ﭘﻴﺸﻨﻬﺎﺩ ﺩﺭ ﺗﻤﺎﻡ ﻣﻮﻟﻔﻪﻫﺎ‬
‫ﺑﺎ ﻫﻢ ﺑﺮﺍﺑﺮﻧﺪ ﻭ ﺗﻨﻬﺎ ﻣﻤﮑﻦ ﺍﺳﺖ ﺩﺭ ﻣﻮﻟﻔﻪﻫﺎﻳﻲ ﮐﻪ ﺑﺎﺯﻳﮑﻦ‪،‬‬
‫ﻋﻀﻮ ‪ S‬ﺍﺳﺖ ﺍﺧﺘﻼﻑ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ‪ .‬ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ ﭘﻴﺸﻨﻬﺎﺩ‬
‫ﺁﻥ ﺑﺎﺯﻳﮑﻦ ﺩﺭ ’‪ u‬ﺧﻼﻑ ﻭﺍﻗﻊ ﻭ ﺑﺮﺍﻱ ﻣﻨﻔﻌﺖ ﺑﻴﺸﺘﺮ ﺍﺳﺖ‪ .‬ﺑﺎ‬
‫ﺍﻳﻦ ﭘﻴﺸﻨﻬﺎﺩﺍﺕ‪ ،‬ﻣﮑﺎﻧﻴﺰﻡ‪ ،‬ﺩﻭ ﺧﺮﻭﺟﻲ )‪ (Q, p‬ﻭ )’‪ (Q’, p‬ﺭﺍ‬
‫ﺑﺪﺳﺖ ﻣﻲﺩﻫﺪ‪ .‬ﻣﻲﮔﻮﻳﻴﻢ ﻳﮏ ﻣﮑﺎﻧﻴﺰﻡ ﮔﺮﻭﻫﻲ ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ‬
‫ﺍﺳﺖ ﺍﮔﺮ ﺍﻳﻦ ﻃﻮﺭ ﻧﺒﺎﺷﺪ ﮐﻪ ﭘﻴﺸﻨﻬﺎﺩ ﺧﻼﻑ ﻭﺍﻗﻊ ﻳﮏ ﮔﺮﻭﻩ‬
‫ﺑﺎﻋﺚ ﺑﺪﺗﺮ ﺷﺪﻥ ﺳﻮﺩ ﻫﻴﭻ ﺑﺎﺯﻳﮑﻨﻲ ﺍﺯ ﺁﻥ ﻧﺸﻮﺩ ﻭ ﺣﺪﺍﻗﻞ‬
‫ﻳﮏ ﻧﻔﺮ ﺳﻮﺩ ﺑﻴﺸﺘﺮﻱ ﻧﻴﺰ ﺑﺒﺮﺩ‪ .‬ﺑﻪ ﻋﺒﺎﺭﺕ ﺭﺳﻤﻲ‪:‬‬
‫∙‬
‫∈ ∀ ∙ ⊆ ∀(‬
‫≥‬
‫⇒ ‪−‬‬
‫=‬
‫) ‪−‬‬
‫⇒‬
‫‪−‬‬
‫‪−‬‬
‫‪ℎ‬‬
‫‪ℳ‬‬
‫‪ Moulin‬ﻧﺸﺎﻥ ﺩﺍﺩ ﮐﻪ ﺭﻭﺵﻫﺎﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪﺍﻱ ﮐﻪ‬
‫ﻳﮑﻨﻮﺍﻳﻲ ﺑﻴﻦﮔﺮﻭﻫﻲ ﺭﺍ ﭘﺸﺘﻴﺒﺎﻧﻲ ﮐﻨﻨﺪ ﻣﻲﺗﻮﺍﻧﻨﺪ ﺩﺭ ﻃﺮﺍﺣﻲ‬
‫ﻣﮑﺎﻧﻴﺰﻡﻫﺎﻱ ﮔﺮﻭﻫﻲ ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﻣﻮﺭﺩ ﺍﺳﺘﻔﺎﺩﻩ ﻗﺮﺍﺭ ﮔﻴﺮﻧﺪ‪.‬‬
‫ﻣﺮﺑﻮﻁ ﺭﺍ ﻧﺴﺒﺖ ﻣﻲﺩﻫﺪ‪ .‬ﺍﻳﻦ ﺭﻭﻳﻪ ﺭﺍ ﻣﺘﻮﺍﺯﻥ ﺑﺎ ﺿﺮﻳﺐ‬
‫ﻣﻲﮔﻮﻳﻴﻢ ﺍﮔﺮ ﻣﺠﻤﻮﻉ ﺗﻤﺎﻡ ﺑﻮﺩﺟﻪ ﺍﺧﺘﺼﺎﺻﻲ ﺑﻪ ﺍﻓﺮﺍﺩ ﻳﮏ‬
‫ﻣﺠﻤﻮﻋﻪ ﺍﺯ ﻫﺰﻳﻨﻪ ﺁﻥ ﻣﺠﻤﻮﻋﻪ ﺑﻴﺸﺘﺮ ﻭ ﺍﺯ ﺑﺮﺍﺑﺮ ﺁﻥ ﮐﻤﺘﺮ‬
‫ﻧﺒﺎﺷﺪ‪ .‬ﺑﻪ ﺻﻮﺭﺕ ﺭﺳﻤﻲ‪ ،‬ﺭﻭﻳﻪ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺑﻪ ﻓﺮﻡ‬
‫‪ ξ: × ℙ → ℝ‬ﺍﺳﺖ ﺑﻪ ﻃﻮﺭﻱ ﮐﻪ‬
‫‪∙ ξ( , ) = 0‬‬
‫∉ ∀∙‬
‫⊆ ∀‬
‫ﺗﻌﺮﻳﻒ‪ :۴‬ﻳﮏ ﺭﻭﻳﻪ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ‪ ξ‬ﺑﻴﻦﮔﺮﻭﻫﻲ ﻳﮑﻨﻮﺍ ﺍﺳﺖ‬
‫ﺍﮔﺮ‬
‫) ∪ ‪∙ (, )≥ (,‬‬
‫∈ ∀∙‬
‫⊆‬
‫‪∀ ,‬‬
‫ﺩﺭ ﺯﻳﺮ ﻣﻲﺑﻴﻨﻴﻢ ﮐﻪ ﻳﮑﻨﻮﺍﻳﻲ ﺑﻴﻦﮔﺮﻭﻫﻲ ﻭﻳﮋﮔﻲ ﻗﻮﻱﺗﺮﻱ‬
‫ﻧﺴﺒﺖ ﺑﻪ ﻫﺴﺘﻪ ﺑﺎﺯﻱ ﺍﺳﺖ‪.‬‬
‫ﻗﻀﻴﻪ‪ :۳‬ﺍﮔﺮ ﻳﮏ ﺭﻭﻳﻪ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ‪ ،‬ﺑﻴﻦﮔﺮﻭﻫﻲ ﻳﮑﻨﻮﺍ ﻭ‬
‫ﺩﺍﺭﺍﻱ ﺗﻮﺍﺯﻥ ﭘﻮﻟﻲ ﺑﺎ ﺿﺮﻳﺐ ﺑﺎﺷﺪ‪ ،‬ﺁﻧﮕﺎﻩ ﻣﻘﺎﺩﻳﺮ ﺁﻥ ﺩﺭ‬
‫ﻫﺴﺘﻪ‬
‫ﺑﺎﺯﻱ ﺍﺳﺖ‪.‬‬
‫ﺍﺛﺒﺎﺕ‪ :‬ﺗﻨﻬﺎ ﮐﺎﻓﻲ ﺍﺳﺖ ﮐﻪ ﺍﺛﺒﺎﺕ ﮐﻨﻴﻢ ﺭﻭﻳﻪ ﺗﻮﺯﻳﻊ ﺩﺍﺭﺍﻱ‬
‫ﻭﻳﮋﮔﻲ ﻫﺴﺘﻪ ﺍﺳﺖ‪ .‬ﻳﻌﻨﻲ‬
‫) ( ≤) ‪(,‬‬
‫∈‬
‫∙‬
‫⊆ ∀‬
‫ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻳﮑﻨﻮﺍﻳﻲ ﺑﻴﻦﮔﺮﻭﻫﻲ ﺩﺍﺭﻳﻢ ﮐﻪ‬
‫) ‪∙ (, )≤ (,‬‬
‫∈ ∀‬
‫) ‪(,‬‬
‫∈‬
‫⇒) ‪∙ (, )≤ (,‬‬
‫) ( ≤) ‪( ,‬‬
‫∈‬
‫≤‬
‫∈ ∀‬
‫ﻧﺎﻣﺴﺎﻭﻱ ﺁﺧﺮ ﺍﺯ ﺧﺎﺻﻴﺖ ﺗﻮﺍﺯﻥ ﭘﻮﻟﻲ ﺑﺪﺳﺖ ﻣﻲﺁﻳﺪ‪.‬‬
‫ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻳﻦ ﺭﻭﻳﻪ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ‪ ،‬ﻣﮑﺎﻧﻴﺰﻡ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ‬
‫‪ ℳ‬ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺯﻳﺮ ﺗﻌﺮﻳﻒ ﻣﻲﮐﻨﻴﻢ‪.‬‬
‫‪Mechanism ℳ‬‬
‫← ‪Initialize‬‬
‫‪Repeat‬‬
‫}) ‪← { ∈ | ≥ ( ,‬‬
‫) ‪Until for all ∈ ∙ ≥ ( ,‬‬
‫‪Return‬‬
‫=‬
‫‪= ( , ) for all i‬‬
‫ﻳﮏ ﺭﻭﻳﻪ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺑﻴﻦﮔﺮﻭﻫﻲ ﻳﮑﻨﻮﺍ‬
‫ﻗﻀﻴﻪ‪ :۴‬ﺍﮔﺮ‬
‫ﺑﺎﺷﺪ ﻭ }‪∈ ℝ+ ∪ {0‬‬
‫ﺟﺰﻭ‬
‫∗‬
‫ﻧﺨﻮﺍﻫﺪ ﺑﻮﺩ ﮐﻪ ﻳﮏ ﺗﻨﺎﻗﺾ ﺍﺳﺖ‪.‬‬
‫ﻗﻀﻴﻪ ﺯﻳﺮ ﺑﻴﺎﻥ ﻣﻲﮐﻨﺪ ﮐﻪ‬
‫ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﺍﺳﺖ‪.‬‬
‫ﻗﻀﻴﻪ‪ :۵‬ﺍﮔﺮ‬
‫‪ ℳ‬ﻳﮏ ﻣﮑﺎﻧﻴﺰﻡ ﮔﺮﻭﻫﻲ‬
‫ﻳﮏ ﺭﻭﻳﻪ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺑﻴﻦﮔﺮﻭﻫﻲ ﻳﮑﻨﻮﺍ ﺑﺎ‬
‫ﺗﻮﺍﺯﻥ ﭘﻮﻟﻲ ﺑﻪ ﺿﺮﻳﺐ‬
‫ﺑﺎﺷﺪ‪ ،‬ﺁﻧﮕﺎﻩ‬
‫ﮔﺮﻭﻫﻲ ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﺑﺎ ﺗﻮﺍﺯﻥ ﭘﻮﻟﻲ ﺍﺳﺖ‪.‬‬
‫‪ ℳ‬ﻳﮏ ﻣﮑﺎﻧﻴﺰﻡ‬
‫ﻭﺟﻮﺩ ﺩﺍﺭﺩ‬
‫‪ .‬ﻣﮑﺎﻧﻴﺰﻡ ‪ℳ‬‬
‫ﻏﻴﺮﻭﺍﻗﻌﻲ ’‪ u‬ﺑﻪ ﺟﺎﻱ ‪ u‬ﻣﻨﻔﻌﺖ ﺑﻴﺸﺘﺮﻱ ﮐﺴﺐ ﮐﻨﺪ‪ .‬ﻓﺮﺽ‬
‫⊆‬
‫ﮐﻪ ﺗﻮﺳﻂ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺑﺎﻻ ﺑﺪﺳﺖ ﻣﻲﺁﻳﺪ ﻳﮑﺘﺎ‬
‫ﻭﺟﻮﺩ ﺧﻮﺍﻫﺪ ﺩﺍﺷﺖ ﮐﻪ ﺁﻥ ﻧﻴﺰ ﺑﻴﺸﻴﻨﻪ‬
‫ﺍﺳﺖ ﻭ ﺑﺮﺍﻱ ﻫﺮ ﻋﻀﻮ ‪ i‬ﺁﻥ ) ‪≥ ( ,‬‬
‫‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﻃﺒﻖ‬
‫) ∪ ‪≥ (,‬‬
‫ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪:‬‬
‫ﮐﻨﻴﺪ ﮐﻪ ‪ i‬ﺑﺎ ﺍﻳﻦ ﮐﺎﺭ ﮔﺮﻭﻩ‪ ،‬ﺑﺮﻧﺪﻩ ﺷﻮﺩ‪ .‬ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ‬
‫≥ ) ‪ . ′ ≥ ( ,‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﻫﺰﻳﻨﻪﺍﻱ ﮐﻪ ﺷﺨﺺ‬
‫ﭘﺮﺩﺍﺧﺖ ﻣﻲﮐﻨﺪ ﺍﺯ ﺍﺭﺯﺷﻲ ﮐﻪ ﺧﺪﻣﺎﺕ ﻣﻮﺭﺩ ﻧﻈﺮ ﺑﺮﺍﻳﺶ‬
‫ﺩﺍﺭﺩ ﺑﻴﺸﺘﺮ ﺍﺳﺖ ﮐﻪ ﺑﻪ ﻧﻔﻊ ﺍﻭ ﻧﻴﺴﺖ‪.‬‬
‫ﺍﺯ ﻃﺮﻑ ﺩﻳﮕﺮ ﭼﻮﻥ ﭘﻴﺸﻨﻬﺎﺩ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺑﺮﻧﺪﻩ ﺩﺭ ﺳﻨﺎﺭﻳﻮﻱ‬
‫ﻳﮑﻨﻮﺍﻳﻲ ﺧﻮﺍﻫﻴﻢ ﺩﺍﺷﺖ‪:‬‬
‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻣﺠﻤﻮﻋﻪ‬
‫ﻧﺎﻣﺴﺎﻭﻱ ﺍﺧﻴﺮ‪،‬‬
‫)∗‬
‫‪< (,‬‬
‫ﺑﺪﺳﺖ ﻣﻲﺁﻳﺪ‪ .‬ﭘﺲ ‪i‬‬
‫ﮔﺮﻭﻫﻲ ﻣﺎﻧﻨﺪ ‪ T‬ﻭﺟﻮﺩ ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ ﮐﻪ ﺑﺎ ﺍﻋﻼﻡ ﭘﻴﺸﻨﻬﺎﺩ‬
‫ﺍﻳﻦ ﻣﺠﻤﻮﻋﻪ ﺭﺍ ﺑﺪﺳﺖ ﻣﻲﺩﻫﺪ‪.‬‬
‫ﻫﻤﻴﻦﻃﻮﺭ ﺑﺮﺍﻱ‬
‫ﻳﮑﻨﻮﺍﻳﻲ ﺑﻴﻦﮔﺮﻭﻫﻲ‬
‫)∗‬
‫‪ . ( , ) < ( ,‬ﺑﻨﺎﺑﺮ ﺩﻭ‬
‫ﭘﻴﺸﻨﻬﺎﺩ ﻗﻴﻤﺖ ﺑﺮﺍﻱ ﺑﺎﺯﻳﮑﻦ ‪i‬ﺍﻡ‬
‫ﮐﻪ ﺑﺮﺍﻱ ﻫﺮ ﻋﻀﻮ ‪ i‬ﺁﻥ ) ‪≥ ( ,‬‬
‫ﻧﺒﺎﺷﺪ‪ ،‬ﺁﻧﮕﺎﻩ‬
‫ﻣﺠﻤﻮﻋﻪ ﻣﺠﻤﻮﻋﻪ ﻧﻬﺎﻳﻲ ﺑﻴﺸﻴﻨﻪ )‬
‫∗‬
‫( ﺍﺳﺖ‪ .‬ﭘﺲ ﻃﺒﻖ‬
‫ﺍﺛﺒﺎﺕ‪ :‬ﺍﺯ ﺑﺮﻫﺎﻥ ﺧﻠﻒ ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﮐﻨﻴﻢ‪ .‬ﻓﺮﺽ ﮐﻨﻴﺪ ﮐﻪ‬
‫ﺑﺎﺷﺪ‪ ،‬ﺁﻧﮕﺎﻩ ﻳﮏ ﻣﺠﻤﻮﻋﻪ ﺑﻴﺸﻴﻨﻪ ﻭ ﻳﮑﺘﺎ‬
‫ﺍﺛﺒﺎﺕ‪ :‬ﺍﮔﺮ‬
‫ﺭﺍ ﻫﺮﺑﺎﺭ ﮐﻮﭼﮏ ﻣﻲﮐﻨﺪ‪ ،‬ﻣﻲﺗﻮﺍﻧﻴﻢ ﻧﺘﻴﺠﻪ ﺑﮕﻴﺮﻳﻢ ﮐﻪ‬
‫ﺯﻳﺮ‬
‫∙‬
‫∈ ∀‬
‫) ∪ ‪∀ ∈ ∙ ≥ (,‬‬
‫∪ ﭼﻮﻥ ﺍﺯ ﺩﻭ ﻣﺠﻤﻮﻋﻪ ﺩﻳﮕﺮ‬
‫ﺑﺰﺭﮔﺘﺮ ﺍﺳﺖ ﺑﺎﻳﺪ ﺑﻴﺸﻴﻨﻪ ﺑﺎﺷﺪ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﻓﺮﺽ ﺍﻭﻝ ﮐﻪ ﺩﻭ‬
‫ﻣﺠﻤﻮﻋﻪ ﺑﻴﺸﻴﻨﻪ ﺩﺍﺷﺘﻴﻢ ﺍﺷﺘﺒﺎﻩ ﺑﻮﺩ‪.‬‬
‫ﺣﺎﻻ ﺍﺛﺒﺎﺕ ﻣﻲﮐﻨﻴﻢ ﮐﻪ ﺍﻟﮕﻮﺭﻳﺘﻢ ﺑﺎﻻ ﺗﻤﺎﻡ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺣﺎﺋﺰ‬
‫ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﮐﻤﺘﺮ ﺍﺯ ﭘﻴﺸﻨﻬﺎﺩ ﺁﻥ ﺩﺭ ﺳﻨﺎﺭﻳﻮﻱ ﻏﻴﺮﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ‬
‫ﻧﻴﺴﺖ‪،‬‬
‫ﺑﻴﺸﻴﻨﻪ ﺍﺳﺖ ﻭ ﺩﺍﺭﻳﻢ‬
‫⊆ ‪ . ′‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﺑﺮ ﺍﺳﺎﺱ‬
‫ﻳﮑﻨﻮﺍﻳﻲ ﺑﻴﻦﮔﺮﻭﻫﻲ‪ ،‬ﻫﺰﻳﻨﻪ ﻫﺮ ﺷﺨﺺ ﺩﺭ ﺳﻨﺎﺭﻳﻮﻱ‬
‫ﻏﻴﺮﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﺣﺪﺍﻗﻞ ﺑﻪ ﺍﻧﺪﺍﺯﻩ ﻫﺰﻳﻨﻪﺍﺵ ﺩﺭ ﺳﻨﺎﺭﻳﻮﻱ‬
‫ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﺍﺳﺖ‪ .‬ﺑﻨﺎﺑﺮﺍﻳﻦ ﻫﻴﭻ ﮐﺴﻲ ﺩﺭ ﺳﻨﺎﺭﻳﻮﻱ‬
‫ﻏﻴﺮﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﺳﻮﺩ ﺑﻴﺸﺘﺮﻱ ﻧﻤﻲﺑﺮﺩ‪.‬‬
‫ﺷﺮﺍﻳﻂ ﺭﺍ ﺩﺭ ﻣﺠﻤﻮﻋﻪ ﺑﻴﺸﻴﻨﻪ ﻭﺍﺭﺩ ﻣﻲﮐﻨﺪ‪ .‬ﺍﺯ ﺑﺮﻫﺎﻥ ﺧﻠﻒ‬
‫‪ -۵‬ﻣﻘﺪﺍﺭ ‪Shapley‬‬
‫ﺍﺳﺘﻔﺎﺩﻩ ﻣﻲﮐﻨﻴﻢ‪ .‬ﻓﺮﺽ ﮐﻨﻴﺪ ﮐﻪ ﺑﺎﺯﻳﮑﻦ ‪ ،i‬ﺍﻭﻟﻴﻦ ﺑﺎﺯﻳﮑﻨﻲ‬
‫ﻳﮑﻲ ﺍﺯ ﻣﺸﮑﻼﺕ ﻫﺴﺘﻪ ﺩﺭ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺍﻳﻦ ﺍﺳﺖ‬
‫ﺍﺳﺖ ﮐﻪ ﺣﺬﻑ ﻣﻲﺷﻮﺩ ﺩﺭ ﺻﻮﺭﺗﻲ ﮐﻪ ﻧﺒﺎﻳﺪ ﺣﺬﻑ ﻣﻲﺷﺪ‪.‬‬
‫ﮐﻪ ﺑﻪ ﻧﺪﺭﺕ ﻳﮏ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﻳﮑﺘﺎ ﺑﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻧﺴﺒﺖ ﺩﺍﺩﻩ‬
‫‪ .‬ﺍﺯ ﻃﺮﻓﻲ ﭼﻮﻥ‬
‫ﻣﻲﺷﻮﺩ‪ .‬ﻣﻌﻤﻮﻻ ﻳﺎ ﻫﺴﺘﻪ ﺗﻬﻲ ﺍﺳﺖ ﻭ ﻳﺎ ﺍﻳﻨﮑﻪ ﭼﻨﺪ ﻧﻘﻄﻪ ﺩﺭ‬
‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﻃﺒﻖ ﺍﻟﮕﻮﺭﻳﺘﻢ ) ‪< ( ,‬‬
‫ﺍﻟﮕﻮﺭﻳﺘﻢ ﺑﺮﺍﻱ ﺑﺪﺳﺖ ﺁﻭﺭﺩﻥ ﻣﺠﻤﻮﻋﻪ ﺑﻴﺸﻴﻨﻪ‪ ،‬ﻣﺠﻤﻮﻋﻪ‬
‫ﺁﻥ ﻗﺮﺍﺭ ﺩﺍﺭﺩ ﮐﻪ ﻃﺮﺍﺡ ﺭﺍ ﻣﺠﺒﻮﺭ ﻣﻲﮐﻨﺪ ﺍﺯ ﺷﺮﺍﻳﻂ ﺩﻳﮕﺮ‬
‫ﺑﺮﺍﻱ ﻣﺤﺪﻭﺩ ﮐﺮﺩﻥ ﺁﻥ ﺍﺳﺘﻔﺎﺩﻩ ﮐﻨﺪ‪ Shapley .‬ﺗﻮﺯﻳﻊ‬
‫‪ submodular‬ﻫﺮ ﺭﻭﺵ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺍﻓﺰﺍﻳﺸﻲ ﮐﻪ ﻣﻘﺪﺍﺭ‬
‫ﻫﺰﻳﻨﻪﺍﻱ ﺭﺍ ﻣﻌﺮﻓﻲ ﮐﺮﺩ ﮐﻪ ﺑﻪ ﺻﻮﺭﺕ ﻳﮑﺘﺎ ﻭ ﺑﺎ ﺷﺮﺍﻳﻂ‬
‫‪ Shapley‬ﺭﺍ ﺷﺎﻣﻞ ﻣﻲﺷﻮﺩ‪ ،‬ﺩﺭ ﻫﺴﺘﻪ ﺑﺎﺯﻱ ﻗﺮﺍﺭ ﺩﺍﺭﻧﺪ‪.‬‬
‫ﻋﺎﺩﻻﻧﻪ ﻫﺰﻳﻨﻪ ﺭﺍ ﺗﻮﺯﻳﻊ ﻣﻲﮐﻨﺪ‪.‬‬
‫ﺩﺭ ﻭﺍﻗﻊ ﻣﻘﺪﺍﺭ ‪ Shapley‬ﻳﮏ ﺭﻭﺵ ﺑﻴﻦﮔﺮﻭﻫﻲ ﻳﮑﻨﻮﺍ ﺍﺳﺖ‬
‫ﺑﺎﺯﻱ )‪ (A, c‬ﺭﺍ ﺩﺭ ﻧﻈﺮ ﻣﻲﮔﻴﺮﻳﻢ‪ .‬ﻳﮏ ﺭﺍﻩ ﺳﺎﺩﻩ ﺗﻮﺯﻳﻊ‬
‫ﮐﻪ ﻣﻲﺗﻮﺍﻧﺪ ﺩﺭ ﻃﺮﺍﺣﻲ ﻣﮑﺎﻧﻴﺰﻡﻫﺎﻱ ﮔﺮﻭﻫﻲ ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ‬
‫ﻫﺰﻳﻨﻪ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺭﺍ ﺑﻪ ﺗﺮﺗﻴﺒﻲ ﻣﺮﺗﺐ ﮐﻨﻴﻢ ﻭ‬
‫ﻣﻮﺭﺩ ﺍﺳﺘﻔﺎﺩﻩ ﻗﺮﺍﺭ ﮔﻴﺮﺩ‪.‬‬
‫ﺳﭙﺲ ﺑﻪ ﺗﺮﺗﻴﺐ ﻟﻴﺴﺖ‪ ،‬ﺑﻪ ﻫﺮﮐﺲ ﻫﺰﻳﻨﻪﺍﻱ ﺭﺍ ﺗﺨﺼﻴﺺ‬
‫ﺩﺭ ﻳﮏ ﺑﺎﺯﻱ ‪ ،submodular‬ﺑﺮﺍﻱ ﻫﺮ ﺩﻭ ﺯﻳﺮﻣﺠﻤﻮﻋﻪﺍﻱ ﺍﺯ‬
‫ﺑﺪﻫﻴﻢ ﮐﻪ ﺍﺯ ﺍﺿﺎﻓﻪ ﺷﺪﻥ ﺍﻭ ﺑﻪ ﮔﺮﻭﻩ ﺑﺪﺳﺖ ﻣﻲﺁﻳﺪ‪ .‬ﺑﻪ ﺍﻳﻦ‬
‫ﺑﺎﺯﻳﮑﻨﺎﻥ ﻣﺎﻧﻨﺪ‬
‫) ( ‪) ≤ ( )+‬‬
‫ﻫﺰﻳﻨﻪ‪ ،‬ﻫﺰﻳﻨﻪ ﻣﺮﺯﻱ ﻣﻲﮔﻮﻳﻨﺪ‪ .‬ﺑﻪ ﻋﺒﺎﺭﺕ ﺩﻳﮕﺮ ﺑﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ‬
‫‪ i‬ﺑﻪ ﺍﻧﺪﺍﺯﻩ) ( ‪ ( ∪ { }) −‬ﻫﺰﻳﻨﻪ ﺍﺧﺘﺼﺎﺹ ﻣﻲﻳﺎﺑﺪ‪.‬‬
‫ﺑﻪ ﺍﻳﻦ ﺭﻭﺵ‪ ،‬ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺍﻓﺰﺍﻳﺸﻲ ﻣﻲﮔﻮﻳﻨﺪ‪.‬‬
‫ﺧﻮﺍﺹ ﻣﻘﺪﺍﺭ ‪Shapley‬‬
‫·‬
‫ﻣﺸﮑﻞ ﺭﻭﺵ ﺑﺎﻻ ﺩﺭ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﺑﻲﻧﺎﻡ ﻧﻴﺴﺖ‪ .‬ﻳﻌﻨﻲ ﺗﺮﺗﻴﺐ‬
‫ﺑﺎﺯﻳﮑﻨﺎﻥ ﺩﺭ ﺁﻥ ﻣﻬﻢ ﺍﺳﺖ‪ .‬ﺑﺮﺍﻱ ﺭﻓﻊ ﺍﻳﻦ ﻧﻘﺺ‪Shapley ،‬‬
‫ﺑﻲﻧﺎﻡ‪ :‬ﺗﻐﻴﻴﺮ ﻧﺎﻡ )ﺗﺮﺗﻴﺐ( ﺑﺎﺯﻳﮑﻨﺎﻥ ﺑﺮ ﻣﻘﺪﺍﺭ ﻫﺰﻳﻨﻪ ﺁﻧﻬﺎ‬
‫·‬
‫ﺑﺎﺯﻳﮑﻦ ﺑﻲﺗﺎﺛﻴﺮ‪ :‬ﺍﮔﺮ ﺑﺎﺯﻳﮑﻨﻲ ﻫﺰﻳﻨﻪﺍﻱ ﺭﺍ ﺗﺤﻤﻴﻞ‬
‫ﻧﻤﻲﮐﻨﺪ‪ ،‬ﻧﺒﺎﻳﺪ ﻫﺰﻳﻨﻪﺍﻱ ﺭﺍ ﻧﻴﺰ ﺑﭙﺮﺩﺍﺯﺩ‪.‬‬
‫·‬
‫ﺍﻓﺰﻭﺩﻧﻲ‪ :‬ﻣﻘﺪﺍﺭ ‪ Shapley‬ﻣﺠﻤﻮﻉ ﺩﻭ ﺗﺎﺑﻊ ﻫﺰﻳﻨﻪ‬
‫ﻳﮑﺘﺎ ﻭ ﻋﺎﺩﻻﻧﻪ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺖ‪ .‬ﺍﻳﻦ ﮐﺎﺭ ﻣﺎﻧﻨﺪ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ‬
‫ﻣﮑﺎﻥ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻦ ﺑﺎﺯﻳﮑﻦ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺗﺼﺎﺩﻓﻲ ﺩﺭ ﻧﻈﺮ‬
‫ﺑﮕﻴﺮﻳﻢ ﮐﻪ ﺑﺎﻋﺚ ﻣﻲﺷﻮﺩ ﺩﻳﮕﺮ ﺗﺮﺗﻴﺐ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻣﻬﻢ ﻧﺒﺎﺷﺪ‪.‬‬
‫ﺍﺣﺘﻤﺎﻝ ﺍﻳﻨﮑﻪ ﻣﺠﻤﻮﻋﻪ‬
‫ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ﮐﻪ ﺷﺎﻣﻞ ﻳﮏ ﺑﺎﺯﻳﮑﻦ‬
‫‪ i‬ﻧﻴﺴﺖ‪ ،‬ﻗﺒﻞ ﺍﺯ ﺍﻳﻦ ﺑﺎﺯﻳﮑﻦ ﺩﺭ ﺗﺮﺗﻴﺐ ﻗﺮﺍﺭ ﺑﮕﻴﺮﺩ‪ ،‬ﺑﺮﺍﺑﺮ‬
‫ﺍﺳﺖ ﺑﺎ‬
‫!)‬
‫!‬
‫(!‬
‫ﮐﻪ ﺩﺭ ﺁﻥ | | = ‪.‬‬
‫ﺑﻨﺎﺑﺮﺍﻳﻦ ﺍﻣﻴﺪ ﻫﺰﻳﻨﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺑﺮﺍﺑﺮ ﺧﻮﺍﻫﺪ ﺑﻮﺩ ﺑﺎ ﻣﺠﻤﻮﻉ‬
‫ﻫﺰﻳﻨﻪﻫﺎﻱ ﻣﺮﺯﻱ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺩﺭ ﺍﺣﺘﻤﺎﻝ ﺣﻀﻮﺭ ﺁﻥ ﺑﺎﺯﻳﮑﻦ‬
‫ﺩﺭ ﺁﻥ ﻣﺮﺯ‪.‬‬
‫) ( ‪( ∪ { }) −‬‬
‫=) (‬
‫| |‪⊆ \{ }:‬‬
‫!) ‪! ( − 1 −‬‬
‫!‬
‫∩‬
‫( ‪)+‬‬
‫∪‬
‫(‬
‫ﺗﺎﺛﻴﺮ ﻧﺪﺍﺭﺩ‪.‬‬
‫ﺍﻣﻴﺪ ﻫﺰﻳﻨﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺩﺭ ﺗﻤﺎﻡ ﺣﺎﻻﺕﻫﺎﻱ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻦ ﺍﻭ ﺭﺍ‬
‫ﺣﺴﺎﺏ ﮐﺮﺩ ﻭ ﻣﻘﺪﺍﺭ ﺑﺪﺳﺖ ﺁﻣﺪﻩ ﺭﺍ ﺑﻪ ﻋﻨﻮﺍﻥ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ‬
‫ﻭ‬
‫ﺧﺼﻮﺻﻴﺖ ﺯﻳﺮ ﺑﺮﻗﺮﺍﺭ ﺍﺳﺖ‪:‬‬
‫ﻭ‬
‫ﺍﺯ ﺟﻤﻊ ﻣﻘﺪﺍﺭ ‪ Shapley‬ﻫﺮ ﻳﮏ ﺍﺯ ﺁﻧﻬﺎ ﺑﺪﺳﺖ‬
‫ﻣﻲﺁﻳﺪ‪.‬‬
‫ﺑﺮﺍﻱ ﻣﻘﺪﺍﺭ ‪ Shapley‬ﺍﺛﺒﺎﺕ ﻣﻲﺷﻮﺩ ﮐﻪ ﺗﻨﻬﺎ ﺗﻮﺯﻳﻌﻲ ﺍﺳﺖ‬
‫ﮐﻪ ﺧﺼﻮﺻﻴﺖﻫﺎﻱ ﺑﺎﻻ ﺭﺍ ﺩﺍﺭﺍ ﻣﻲﺑﺎﺷﺪ‪.‬‬
‫‪ -۶‬ﭼﺎﻧﻪﺯﻧﻲ ﻧﺶ‬
‫ﻣﺴﺎﻟﻪ ﺩﻳﮕﺮﻱ ﮐﻪ ﺩﺭ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪﺍﻱ ﻣﻄﺮﺡ‬
‫ﻣﻲﺷﻮﺩ‪ ،‬ﺑﺤﺚ ﻣﺬﺍﮐﺮﻩ ﻭ ﭼﺎﻧﻪﺯﻧﻲ ﺍﺳﺖ‪ .‬ﺩﺭ ﻣﺴﺎﻟﻪ ﭼﺎﻧﻪﺯﻧﻲ‬
‫ﺩﻭ ﻳﺎ ﭼﻨﺪ ﺑﺎﺯﻳﮑﻦ ﺑﺮﺍﻱ ﺭﺳﻴﺪﻥ ﺑﻪ ﻳﮏ ﻧﻈﺮ ﻭﺍﺣﺪ ﺑﺎ ﻳﮑﺪﻳﮕﺮ‬
‫ﻣﺬﺍﮐﺮﻩ ﻣﻲﮐﻨﻨﺪ‪ .‬ﻫﺮ ﻃﺮﻑ ﻣﺬﺍﮐﺮﻩ ﻣﻲﺗﻮﺍﻧﺪ ﻣﺬﺍﮐﺮﻩ ﺭﺍ ﺗﺮﮎ‬
‫ﮐﻨﺪ ﮐﻪ ﺩﺭ ﺍﻳﻦ ﺻﻮﺭﺕ ﺧﺮﻭﺟﻲ ﭼﺎﻧﻪﺯﻧﻲ ﻋﺪﻡ ﺗﻮﺍﻓﻖ‬
‫ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪ .‬ﺑﻪ ﻋﺒﺎﺭﺕ ﺭﺳﻤﻲ‪ ،‬ﻳﮏ ﺑﺎﺯﻱ ﭼﺎﻧﻪﺯﻧﻲ ﺑﺎ ﺩﻭ‬
‫ﺍﻳﻦ ﻣﻘﺪﺍﺭ ﻫﻤﺎﻥ ﻫﺰﻳﻨﻪ ﺍﺧﺘﺼﺎﺹﻳﺎﻓﺘﻪ ﺑﻪ ﻫﺮ ﺑﺎﺯﻳﮑﻦ ﺑﻪ ﺍﺳﻢ‬
‫ﺑﺎﺯﻳﮑﻦ )ﺗﻌﻤﻴﻢ ﺁﻥ ﺑﻪ ﺻﻮﺭﺕ ﻣﺸﺎﺑﻪ ﺍﺳﺖ( ﺑﺎ ﻳﮏ ﻣﺠﻤﻮﻋﻪ‬
‫ﻣﻘﺪﺍﺭ ‪ Shapley‬ﺍﺳﺖ‪ .‬ﻻﺯﻡ ﺑﻪ ﺫﮐﺮ ﺍﺳﺖ ﮐﻪ ﺍﻳﻦ ﻣﻘﺪﺍﺭ ﻟﺰﻭﻣﺎ‬
‫‪⊆ℝ‬‬
‫ﺍﺯ ﻧﻘﺎﻁ ﻗﺎﺑﻞ ﻣﺬﺍﮐﺮﻩ ﺩﺭ ﭼﺎﻧﻪﺯﻧﻲ ﻭ ﻳﮏ ﻧﻘﻄﻪ‬
‫ﺑﻪ ﻋﻨﻮﺍﻥ ﻧﻘﻄﻪ ﻋﺪﻡ ﺗﻮﺍﻓﻖ ﻣﺸﺨﺺ ﻣﻲﺷﻮﺩ‪ .‬ﻫﺮ‬
‫ﺑﺎ ﺍﻳﻦ ﻭﺟﻮﺩ ﺛﺎﺑﺖ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﺑﺮﺍﻱ ﻫﺮ ﺑﺎﺯﻱ‬
‫ﻧﻘﻄﻪ ﺩﺭ‬
‫ﮐﻪ ﻳﮏ ﺧﺮﻭﺟﻲ ﻣﺬﺍﮐﺮﻩ ﻣﻲﺗﻮﺍﻧﺪ ﺑﺎﺷﺪ‪ ،‬ﻣﻨﻔﻌﺖ‬
‫ﺩﺭ ﻫﺴﺘﻪ ﺑﺎﺯﻱ ﻗﺮﺍﺭ ﻧﺪﺍﺭﺩ ﺣﺘﻲ ﺍﮔﺮ ﻫﺴﺘﻪ ﺗﻬﻲ ﻧﺒﺎﺷﺪ‪.‬‬
‫∈‬
‫ﻫﺮﮐﺪﺍﻡ ﺍﺯ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺭﺍ ﺑﻴﺎﻥ ﻣﻲﮐﻨﺪ‪ .‬ﭼﻮﻥ ﺍﺿﺎﻓﻪ ﻳﺎ ﮐﻢ ﮐﺮﺩﻥ‬
‫ﻗﻀﻴﻪ‪ :۶‬ﺑﺮﺍﻱ ﻳﮏ ﺑﺎﺯﻱ ﭼﺎﻧﻪﺯﻧﻲ ﮐﻪ ﺧﻮﺍﺹ ﺑﺎﻻ ﺭﺍ ﺩﺍﺭﺍ‬
‫ﻳﮏ ﻣﻘﺪﺍﺭ ﺑﻪ ﻣﻨﻔﻌﺖ ﺑﺎﺯﻳﮑﻨﺎﻥ‪ ،‬ﺗﺎﺛﻴﺮﻱ ﺩﺭ ﺗﺮﺟﻴﺢ ﺁﻧﻬﺎ ﻧﺴﺒﺖ‬
‫ﺑﺎﺷﺪ ﻳﮏ ﺭﺍﻩﺣﻞ ﻭﺍﺣﺪ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﮐﻪ ﺑﻪ ﻫﺮ ﻣﺠﻤﻮﻋﻪ‬
‫ﺑﻪ ﻳﮑﺪﻳﮕﺮ ﻧﺪﺍﺭﺩ ﻣﻲﺗﻮﺍﻧﻴﻢ ﻧﻘﻄﻪ ﻋﺪﻡ ﺗﻮﺍﻓﻖ ﺭﺍ )‪= (0,0‬‬
‫ﺩﺭ ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ‪ .‬ﻓﺮﺽ ﺑﺮ ﺍﻳﻦ ﺍﺳﺖ ﮐﻪ ﺩﺭ‬
‫ﻫﻤﻴﺸﻪ ﻳﮏ‬
‫ﻧﻘﻄﻪ )‬
‫‪,‬‬
‫ﺑﻴﺸﻴﻨﻪ ﺷﻮﺩ‪.‬‬
‫( ﺭﺍ ﻧﺴﺒﺖ ﻣﻲﺩﻫﺪ ﺑﻪ ﻃﻮﺭﻱ ﮐﻪ‬
‫ﻧﻘﻄﻪ ﻣﺜﺒﺖ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﺗﺎ ﻫﻤﻴﺸﻪ ﺩﻟﻴﻠﻲ ﺑﺮﺍﻱ ﻣﺬﺍﮐﺮﻩ ﻭﺟﻮﺩ‬
‫ﺗﺎﺑﻊ ﺳﻮﺩﻱ ﮐﻪ ﺍﻳﻦ ﻗﻀﻴﻪ ﺍﺭﺍﺋﻪ ﻣﻲﺩﻫﺪ ﻣﺜﺎﻟﻲ ﺍﺯ ﺗﻮﺍﺑﻊ ﺗﺠﻤﻴﻊ‬
‫ﺩﺍﺷﺘﻪ ﺑﺎﺷﺪ‪.‬‬
‫ﺍﺳﺖ ﮐﻪ ﻣﻨﻔﻌﺖ ﺍﻓﺮﺍﺩ ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﻳﮏ ﻋﺪﺩ ﮐﻪ ﻧﺸﺎﻥﺩﻫﻨﺪﻩ‬
‫ﺑﺎ ﺍﻳﻦ ﺗﻌﺮﻳﻒ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ ﮐﻪ ﺍﻧﺘﻘﺎﻝ ﻫﺰﻳﻨﻪ ﺩﺭ ﺁﻧﻬﺎ‬
‫ﻣﻨﻔﻌﺖ ﺟﻤﻌﻲ ﺍﺳﺖ‪ ،‬ﻧﺸﺎﻥ ﻣﻲﺩﻫﺪ‪.‬‬
‫ﻭﺟﻮﺩ ﻧﺪﺍﺭﺩ ﺭﺍ ﻣﻲﺗﻮﺍﻥ ﺑﻪ ﺻﻮﺭﺕ ﻳﮏ ﭼﺎﻧﻪﺯﻧﻲ ﻣﺪﻝ ﮐﺮﺩ‪.‬‬
‫ﺍﻟﺒﺘﻪ ﺑﺎﻳﺪ ﻋﻼﻭﻩ ﺑﺮ ﻋﺪﻡ ﺗﻮﺍﻓﻖ ﺍﻓﺮﺍﺩ‪ ،‬ﻋﺪﻡ ﺗﻮﺍﻓﻖ ﮔﺮﻭﻩﻫﺎ ﺭﺍ‬
‫ﻧﻴﺰ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺖ‪.‬‬
‫ﺑﺎ ﺍﻳﻦ ﻣﻘﺪﻣﻪ ﺑﻪ ﺭﺍﻩﺣﻠﻲ ﻣﻲﭘﺮﺩﺍﺯﻳﻢ ﮐﻪ ﻧﺶ ﺑﺮﺍﻱ ﺣﻞ ﻣﺴﺎﺋﻞ‬
‫ﭼﺎﻧﻪﺯﻧﻲ ﺍﺭﺍﺋﻪ ﮐﺮﺩﻩ ﺍﺳﺖ‪.‬‬
‫ﺗﻌﺮﻳﻒ‪ :۵‬ﻳﮏ ﺭﺍﻩﺣﻞ ﺑﺮﺍﻱ ﺑﺎﺯﻱ ﭼﺎﻧﻪﺯﻧﻲ ﻳﮏ ﺗﺎﺑﻊ ﺍﻧﺘﺨﺎﺏ‬
‫ﺍﺟﺘﻤﺎﻋﻲ ﺍﺳﺖ ﺑﻪ ﻃﻮﺭﻱ ﮐﻪ ﺑﻪ ﻫﺮ ﻣﺠﻤﻮﻋﻪ‬
‫ﮐﻪ ﺩﺍﺭﺍﻱ‬
‫ﺧﺼﻮﺻﻴﺖﻫﺎﻱ ﺑﺎﻻ ﺑﺎﺷﺪ‪ ،‬ﻳﮏ ﻧﻘﻄﻪ ﺍﺯ ﺁﻥ ) ∈ ) ( (‬
‫ﺭﺍ ﺍﺧﺘﺼﺎﺹ ﻣﻲﺩﻫﺪ ﮐﻪ ﺧﺼﻮﺻﻴﺎﺕ ﺯﻳﺮ ﺭﺍ ﺩﺍﺭﺍ ﺑﺎﺷﺪ‪:‬‬
‫·‬
‫ﺑﻬﻴﻨﮕﻲ ﭘﺮﺗﻮ ‪ ( ) : ٩‬ﻳﮏ ﻧﻘﻄﻪ ﺑﻬﻴﻨﻪ ﺍﺳﺖ ﺑﺪﻳﻦ‬
‫ﻣﻌﻨﻲ ﮐﻪ ﻫﻴﭻ ﻧﻘﻄﻪﺍﻱ ﺍﺯ ﻧﻈﺮ ﻫﻴﭻﮐﺲ ﺑﻪ ﺁﻥ ﺗﺮﺟﻴﺢ‬
‫ﻧﺪﺍﺭﺩ‪.‬‬
‫·‬
‫ﺗﻘﺎﺭﻥ‪ :‬ﺍﮔﺮ‬
‫ﻣﺘﻘﺎﺭﻥ ﺑﺎﺷﺪ‪ ،‬ﻣﻮﻟﻔﻪﻫﺎﻱ ) ( ﻫﻢ ﺑﺎ‬
‫ﻫﻢ ﺑﺮﺍﺑﺮ ﺧﻮﺍﻫﻨﺪ ﺑﻮﺩ‪.‬‬
‫·‬
‫ﺍﺳﺘﻘﻼﻝ ﺍﺯ ﻣﻘﻴﺎﺱ‪ :‬ﺍﮔﺮ ﻧﻘﺎﻁ ﻣﻮﺭﺩ ﻣﺬﺍﮐﺮﻩ ﺩﺭ ﺿﺮﻳﺒﻲ‬
‫ﺿﺮﺏ ﺷﻮﻧﺪ‪ ،‬ﻧﻘﻄﻪ ﻣﻮﺭﺩ ﺗﻮﺍﻓﻖ ﻧﻴﺰ ﺩﺭ ﺁﻥ ﺿﺮﻳﺐ‬
‫ﺿﺮﺏ ﺧﻮﺍﻫﺪ ﺷﺪ‪.‬‬
‫·‬
‫ﺍﺳﺘﻘﻼﻝ ﺍﺯ ﺍﻧﺘﺨﺎﺏﻫﺎﻱ ﻧﺎﻣﺮﺑﻮﻁ‪ :‬ﺍﮔﺮ ﻳﮏ ﺯﻳﺮﻣﺠﻤﻮﻋﻪ‬
‫ﺍﺯ ﻧﻘﺎﻁ ﻣﺬﺍﮐﺮﻩ ﺩﺍﺭﺍﻱ ﻧﻘﻄﻪ ﺗﻮﺍﻓﻖ ﻣﺬﺍﮐﺮﻩ ﺑﺎﺷﺪ ﺁﻧﮕﺎﻩ‬
‫ﺁﻥ ﻧﻘﻄﻪ‪ ،‬ﻧﻘﻄﻪ ﺗﻮﺍﻓﻖ ﺯﻳﺮﻣﺠﻤﻮﻋﻪ ﻧﻴﺰ ﺧﻮﺍﻫﺪ ﺑﻮﺩ‪.‬‬
‫‪Pareto Optimality‬‬
‫‪ -۷‬ﺟﻤﻊﺑﻨﺪﻱ‬
‫ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ‪ ،‬ﻣﻔﺎﻫﻴﻢ ﺍﻭﻟﻴﻪ ﺑﺎﺯﻱﻫﺎﻱ ﺗﻌﺎﻣﻠﻲ ﺑﺎ ﺗﻤﺮﮐﺰ ﺑﺮ‬
‫ﺑﺎﺯﻱﻫﺎﻱ ﺑﺎ ﻗﺎﺑﻠﻴﺖ ﺍﻧﺘﻘﺎﻝ ﻫﺰﻳﻨﻪ ﺑﺤﺚ ﺷﺪﻧﺪ‪ .‬ﺩﻳﺪﻳﻢ ﮐﻪ‬
‫ﺳﻮﺍﻝﻫﺎﻱ ﺍﻟﮕﻮﺭﻳﺘﻤﻲ ﻣﺎﻧﻨﺪ ﻣﺤﺎﺳﺒﻪ ﻧﺤﻮﻩ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﺑﺎ‬
‫ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺑﺮﻧﺎﻣﻪﻧﻮﻳﺴﻲ ﺧﻄﻲ ﻣﺴﺎﻟﻪ ﺑﻬﻴﻨﻪﺳﺎﺯﻱ ﻣﺘﻨﺎﻇﺮﺷﺎﻥ‬
‫ﻣﺪﻝ ﻣﻲﺷﻮﻧﺪ‪.‬‬
‫ﻣﻔﻬﻮﻡ ﺩﻳﮕﺮﻱ ﮐﻪ ﺑﻪ ﺁﻥ ﭘﺮﺩﺍﺧﺘﻪ ﺷﺪ‪ ،‬ﻣﻔﻬﻮﻡ ﻫﺴﺘﻪ ﺍﺳﺖ ﮐﻪ‬
‫ﺍﻳﻦ ﺍﻣﮑﺎﻥ ﺭﺍ ﻣﻲﺩﻫﺪ ﮐﻪ ﻫﻤﻪ ﺑﺎﺯﻳﮑﻨﺎﻥ ﺩﺭ ﺑﺎﺯﻱ ﻧﻔﻊ ﺑﺒﺮﻧﺪ ﺑﻪ‬
‫ﻃﻮﺭﻱ ﮐﻪ ﻫﻴﭻﮐﺲ ﺍﺯ ﺍﺋﺘﻼﻑ ﮐﻞ ﻧﺨﻮﺍﻫﺪ ﺟﺪﺍ ﺷﻮﺩ‪ .‬ﻣﻔﻬﻮﻡ‬
‫ﺩﻳﮕﺮ‪ ،‬ﻳﮑﻨﻮﺍﻳﻲ ﺑﻴﻦﮔﺮﻭﻫﻲ ﺍﺳﺖ ﮐﻪ ﺑﺎﻋﺚ ﻣﻲﺷﻮﺩ ﺑﺎﺯﻳﮑﻨﺎﻥ‪،‬‬
‫ﺍﻧﮕﻴﺰﻩﻫﺎﻱ ﻭﺍﻗﻌﻲ ﺧﻮﺩ ﺭﺍ ﺩﺭ ﺑﺎﺯﻱ ﻧﺸﺎﻥ ﺩﻫﻨﺪ ﺗﺎ ﺑﺪﻳﻦ‬
‫ﻭﺳﻴﻠﻪ ﺑﺘﻮﺍﻥ ﻣﮑﺎﻧﻴﺰﻡﻫﺎﻱ ﮔﺮﻭﻫﻲ ﺭﺍﺳﺘﮕﻮﻳﺎﻧﻪ ﻃﺮﺍﺣﻲ ﮐﺮﺩ‪.‬‬
‫ﻣﻘﺪﺍﺭ ‪ Shapley‬ﻧﻴﺰ‪ ،‬ﻳﮏ ﺗﻮﺯﻳﻊ ﻫﺰﻳﻨﻪ ﻳﮑﺘﺎ ﻭ ﻋﺎﺩﻻﻧﻪ ﺭﺍ‬
‫ﺑﺮﺍﻱ ﺑﺎﺯﻳﮑﻨﺎﻥ ﻳﮏ ﺑﺎﺯﻱ ﺑﺪﺳﺖ ﻣﻲﺩﻫﺪ ﺩﺭ ﺣﺎﻟﻲ ﮐﻪ ﻫﺴﺘﻪ‬
‫ﺑﺎﺯﻱ ﺍﺯ ﺍﻳﻦ ﻭﻳﮋﮔﻲ ﺑﺮﺧﻮﺭﺩﺍﺭ ﻧﺒﻮﺩ‪.‬‬
‫ﺩﺭ ﺁﺧﺮ ﻧﻴﺰ ﺩﻳﺪﻳﻢ ﮐﻪ ﺑﺮﺍﻱ ﺑﺎﺯﻱﻫﺎﻱ ﭼﺎﻧﻪﺯﻧﻲ ﮐﻪ ﻃﺮﻑﻫﺎﻱ‬
‫ﺑﺎﺯﻱ ﺑﺮ ﺳﺮ ﺭﺳﻴﺪﻥ ﺑﻪ ﻳﮏ ﺗﻮﺍﻓﻖ ﻣﺬﺍﮐﺮﻩ ﻣﻲﮐﻨﻨﺪ‪ ،‬ﻳﮏ‬
‫ﺭﺍﻩﺣﻞ ﺑﻬﻴﻨﻪ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﮐﻪ ﺁﻥ‪ ،‬ﺑﻴﺸﻴﻨﻪ ﺿﺮﺏ ﻣﻮﻟﻔﻪﻫﺎﻱ‬
‫ﻧﻘﺎﻁ ﻣﺬﺍﮐﺮﻩ ﺍﺳﺖ‪.‬‬
‫ﻣﺮﺟﻊ‬
‫‪9‬‬
‫‪§ N. Nisan, T. Roughgarden, E. Tardos, and‬‬
‫‪V.V. Vazirani, Algorithmic Game Theory,‬‬
‫‪Cambridge University Press, 2007.‬‬