21228

۲۱۲۲۸

كاربرد نظريه نقطه ثابت در نظريه بازي و هم ارزي هاي آنها

كارشناسي ارشد

رياضي محض - آناليز

۱۳۹۵-۱۳۹۸

۱۳۹۸/۴/۳

دكتر رضا سعادتي

دكتر جواد وحيدي

رياضي

ﻧﻈﺮﯾﻪ يﺑﺎزيﭘﯿﺶﺑﯿﻨﯽﻣﯽ ﮐﻨﺪﮐﻪﭼﮕﻮﻧﻪاﻓﺮادﻋﻘﻼﯾﯽﺗﺤﺖﻣﻮﻗﻌﯿﺖ ﻫﺎﯾﯽﮐﻪﺑﻪﻫﻢ واﺑﺴﺘﻪﻫﺴﺘﻨﺪ،رﻓﺘﺎرﻣﯽ ﮐﻨﻨﺪ.ﻫﻤﭽﻨﯿﻦاﺑﺰاررﯾﺎﺿﯽﻗﺪرﺗﻤﻨﺪي،ﺑﺮايﭘﯿﺶﺑﯿﻨﯽﺧﺮوﺟﯽ ﻫﺎي ﯾﮏﺑﺎزي،وﻗﺘﯽﺑﻬﺮه ﻣﻨﺪياﻓﺮادﻣﺘﻀﺎدﻣﯽ ﺑﺎﺷﺪرااراﺋﻪﻣﯽ ﮐﻨﺪ. ﮐﻪاﻏﻠﺐدرﻣﻮﻗﻌﯿﺖ ﻫﺎي اﻗﺘﺼﺎديرخﻣﯽ دﻫﺪ. ﺟﺎنﻧﺶﻧﻈﺮﯾﻪ يﺗﻌﺎدلراﺑﻪﺑﺎزي ﻫﺎﯾﯽﺑﺎﺗﻌﺪادﻣﺘﻨﺎﻫﯽﺑﺎزﯾﮑﻦﺗﻌﻤﯿﻢدادوﻫﻤﭽﻨﯿﻦ اﺛﺒﺎﺗﯽﺑﺮايوﺟﻮدﺗﻌﺎدلﻧﺶدراﺳﺘﺮاﺗﮋي ﻫﺎيﺗﺮﮐﯿﺒﯽﺑﺮايﺑﺎزي ﻫﺎيﻣﺘﻨﺎﻫﯽوﺑﻪﻓﺮم اﺳﺘﺮاﺗﮋﯾﮏاراﺋﻪﮐﺮد. وﺟﻮد ﺗﻌﺎدل در ﺑﺎزي ﻫﺎ ﺑﺎ ﻗﻀﺎﯾﺎي ﻧﻘﻄﻪ ي ﺛﺎﺑﺖ ﻣﺎﻧﻨﺪ ﻗﻀﯿﻪ ي ﻧﻘﻄﻪ ي ﺛﺎﺑﺖ ﺑﺮآور و ﮐﺎﮐﻮﺗﺎﻧﯽارﺗﺒﺎطﺑﺴﯿﺎرﻧﺰدﯾﮑﯽدارد. ﻗﻀﯿﻪ يﻣﯿﻨﯿﻤﺎﮐﺲﯾﮑﯽازﻧﺘﺎﯾﺞﻣﻬﻢدرﻧﻈﺮﯾﻪ يﺑﺎزياﺳﺖ. اﯾﻦﻗﻀﯿﻪوﺟﻮدﯾﮏ اﺳﺘﺮاﺗﮋيراﮐﻪﺑﯿﺸﺘﺮﯾﻦﻣﻘﺪارزﯾﺎنراﮐﻤﯿﻨﻪوﮐﻤﺘﺮﯾﻦﻣﻘﺪارﺳﻮدراﺑﯿﺸﯿﻨﻪﻣﯽ ﮐﻨﺪ،ﻧﺸﺎن ﻣﯽ دﻫﺪوﻫﻤﭽﻨﯿﻦﺑﺎاﺳﺘﻔﺎدهازﻧﺘﺎﯾﺞﻗﻀﯿﻪ يﻣﯿﻨﯿﻤﺎﮐﺲ،ﻣﺤﺎﺳﺒﻪ يﺑﻬﺘﺮﯾﻦاﺳﺘﺮاﺗﮋيﮐﻪ ﻫﺮﺑﺎزﯾﮑﻦاﺗﺨﺎذﻣﯽ ﮐﻨﺪ،اﻣﮑﺎنﭘﺬﯾﺮاﺳﺖ. ﻫﻤﭽﻨﯿﻦﻧﺸﺎنﻣﯽ دﻫﯿﻢﮐﻪﻗﻀﯿﻪ يﻧﻘﻄﻪ يﺛﺎﺑﺖﺑﺮآوروﮐﺎﮐﻮﺗﺎﻧﯽﺑﺎاﺳﺘﻔﺎده يﻣﺴﺘﻘﯿﻢ ازﺗﻌﺎدلﻧﺶﺑﻪدﺳﺖﻣﯽ آﯾﺪ. ﮐﻠﻤﺎتﮐﻠﯿﺪي:ﺗﻌﺎدلﻧﺶ؛ﻗﻀﯿﻪ يﻧﻘﻄﻪﺛﺎﺑﺖﺑﺮآور؛ﻗﻀﯿﻪ يﻧﻘﻄﻪﺛﺎﺑﺖﮐﺎﮐﻮﺗﺎﻧﯽ؛ﻗﻀﯿﻪ ي ﻣﯿﻨﯿﻤﺎﮐﺲ

1398/08/06

ََََApplication of fixed point theory in game theory with equivalence results

10/28/2019 12:00:00 AM

Game theory predicts how rational individuals would behave under interdepen dence. It provides a great mathematical tool to infer outcomes when people have conflicts of interests, which often happens in economics situations. John Nash generalized the notion of an equilibrium to games with finite number of players and also established the existence of at least one mixed strategy, Nash equilib rium for every finite strategic form games. The existence of equilibria in games is closely coupled with fixed point theorems such as the Brouwer fixed point theorem and Kakutani’s fixed point theorem. The minimax theorem is one of the most important results in game theory. The theorem shows that there exists a strategy that both minimizes the maximum loss and maximizes the minimum gain. Moreover, by working backwards from the outcomeoftheminimaxtheorem,itispossibletocalculatethebeststrategytheplayers could take. WealsoshowthattheKakutaniandBrouwerfixedpointtheoremscanbeobtained by directly using the Nash equilibrium theorem. Keywords: Nash equilibrium; Brouwer fixed point theorem; Kakutani fixed point theorem; minimax theorem.