http://www.aimspress.com/journal/Math AIMS Mathematics, 3(1): 44–55 DOI:10.3934/Math.2018.1.44 Received: 07 September 2017 Accepted: 30 November 2017 Published: 31 January 2018 Mini review A survey of critical structures in competitive games Amir Hossein Rashme 1 , Zahra Farhad Touski 2, ∗ and Madjid Eshaghi 1 1 Department of Mathematics, Semnan University, Semnan, Iran 2 Center of Excellence in Nolinear Analysis and Applications, Semnan University, Semnan, Iran * Correspondence: Email: z.farhad61@gmail.com; Tel: +98-918-612-5080; Fax: +98-214-423-0653. Abstract: One of the biggest problems of human society is facing crises. Origins of many crises go back to strategy selection in the relations between human beings. The international community is faced with many crises, such as poverty and lack of development of a large section of human society, global warming, economic crises, the incidence of infectious diseases, the accumulation of weapons of mass destruction, wars, migration, lack of food and clean drinking water are among the crises that threaten international community. Each of these challenges alone would require measures and facilities that in many cases are beyond the limited resources of the international community. In this article, the crises have been discussed, whose origin is relations between human beings. By defining critical points in 2 × 2 games, we provide a mathematical model to detect this type of crises, and then by defining a unique compromise point, we offer solutions for this type of crisis. Sometimes the compromise point corresponds to the Nash equilibrium, and sometimes better than Nash equilibrium. We believe that what is presented in this article can help fill the void. Fixing the vacuum in game theory and optimal use of compromise and critical points leads to the development of cooperation–cooperation strategy in the world. Keywords: critical point; compromise point; Cuban Missile Crisis; cooperation strategy; non-cooperative games Mathematics Subject Classification: 91A02 1. Introduction The international community is faced with many crises, such as poverty and lack of development of a large section of human society, global warming, economic crises, the incidence of infectious diseases, the accumulation of weapons of mass destruction, wars, migration, lack of food and clean drinking water are among the crises that threaten international community. The other considerable