The simplest games (from the perspective of logical structure) are those in which agents have perfect information, meaning that at every point where e

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2 The simplest games (from the perspective of logical structure) are those in which agents have perfect information, meaning that at every point where each agent's strategy tells her to take an action, she knows everything that has happened in the game up to that point. This is so because in such games (as long as the games are finite, that is, terminate after a known number of actions) players and analysts can use a straightforward procedure for predicting outcomes. A player in such a game chooses her first action by considering each series of responses and counter-responses that will result from each action open to her. She then asks herself which of the available final outcomes brings her the highest utility, and chooses the action that starts the chain leading to this outcome. This process is called backward induction (because the reasoning works backwards from eventual outcomes to present choice problems). Nash equilibrium (NE) stages are the set of strategies so that no player can maximize payoff by unilateral deviations. John Forbes Nash devised a theorem for it which are known as Nash equilibrium stages. SNE(strong Nash equilibrium stages are the refinements of Nash equilibrium such that no player is going to benefit by unilateral or bilateral defimitions : , /4:9/ ,8 06: -7 :2 89, :3.9 43,,205, 0/ ,9 02,9., ,3// 8.4; $9743,8 6: -7 :2 $ 89, /4: : -7 :289,90,3/9 : /,/4590/ , / , 078,9 02, :2-078,70 -,8.,,3 :, ,, ,9:70:2-078,702,907, , ,, ,9:70 % ,,850.98, / 9 0, ,9:70 9,9 8

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9 9,9/4349 3;4 ;0 3.70/ , $0.43/ 0; , ,9 43,8,9 0,89430$ :3.9 43, 06:,9 43,20 5, 0/ - # 02,33 09, 1: ,709 45, /8948 3,3/ / % 84 : , , ,8 06: -7 :2 89, ,8 06: -7 :2 89, / :9 43,.,7/ 3/: ,, ,9 43,, 8 0,/ 94,9 0,89 3,8 06: -7 :2 89, 0 Number system as an originally found quantum group: / : , 4., ;, # 02,33 09,1: Number system is a physically FOUND commutative GROUP as in quantum world. a) For every a, b there exists a+b on the number line(group) b) There exists an identity element I su ch that a+i=i+a=a This will also be valid. if operator + is changed to *. Any way ^,* are derived out of + So, with any one operator, the number system is DEFINITELY QUANTUM GROUP FNATURE. On numbr line for everyelement 1) a, there exists 1/a, 2) a there exists a. Thus covering each and every point on the number line. Every a can generate another number b on it through operators +,-,*,/,^ 4)Entire negative part of number line is mapped onto the domain 0-1

10 e.g. 2^-3=1/8,2^-2=1/4,2^-1=1/2 5)and every positive side on number line can be mapped onto the domain after 1 e.g.,2^1=2,2^2=4,2^3=8 etc. 6) 0 is mapped onto 1 to maintain the symmetry of GROUP i.e. 2^0=1 What I am interested here is to so why NUMBER SYSTEM is an INTERCONNECRED GROUP. So, I wd like to say that THIS HAS ALL THE PROPERTY OF A QUANTUM SYMMETRY GROUP FOUND IN NATURE and being a qwuantum group IT S INTERCONCTED and perfectly informed group. Refe. Eugen Merzbacher -Quantum mechanics Chapter 17, Groups and Symmetrical invariancy in ORIGINALLY FOUND GROUP IN nature.!# :3.9 43, 06:,9 438,9 81 0/-!, /42, ,3/ ,. 8 / ,3,3/2: , :, : <, ,

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13 % :8$ 89 0$ 39 8,20-,.,7/ 3/:.9 43 % $ $% 84 :9 43 $89, 0 $$&!! #% & $% #,3/ : / /-, 9 05, , , : 9 8,20,8,8 06: -7 :289,908,83430., ,83430., :3,907, /0;, : ,90 4: /-09 0 $ $%# $ #"& # &89,90, : :3,907, /0;,9 438,3/.4, 9 43/0;,9 4383/ , 0782: / , , 411 % 8, :9 1: # 02, ,997 ;, ,3/97 ;, # 8 % : , # 8 Г 9147 # 8,3/, # 8 " Reference: 1.GAME THEORY 2. Extensive form finite game of perfect information ard.edu/cs286r/archived/fall08/files/lecture3.pdf+finite+extensive+form+ga me&hl=en&gl=in&pid=bl&srcid=adgeesglax41fnxre9yxw1k51fnhuya19bv 8Yf3WmRbHOwAmahc33o2QaSHi2Ns0Q6dk4IUid2avKWibOoXKmC20Xm3Y

14 8c7NlxzmceD2qWQzeNFz4FsjlWIaJbAN4NDC_bq3tBAutI5F&sig=AHIEtbT22 Mjxlsq1 3) Official description of Riemann Hypothesis problem by Enrico Bombeiri Pw1hpwmsJONtCFAKQhttp://docs.google.com/viewer?a=v&q=cache:IdUCngHnmOoJ: math.org/millennium/riemann_hypothesis/riemann.pdf+rimeann+hypothesis+ooficial+descript ion+by+bombeiri&hl=en&gl=in&pid=bl&srcid=adgeesg857veuf9atzsyjwn1bisnb25hoha7m3vuc2irjjrsgs4lbklfgvtijxtsinmmiiy8wuicuih54gkkwy1wjbpfz4fdvk9bfjuqwx o2s4btahyq_beislbqg6gi4cdjohoxaq0&sig=ahietbsa-khwurcachkjvad9zl5b5oj4cq 4) Riemann zeta functional equation in the entire complex plane. n&safe=off&sa=n&gbv=2&ndsp=21&sout=0&tbm=isch&tbnid=5rsllrm5ngh lkm:&imgrefurl= 2005/CourseHome/index.htm&docid=cECkPL8ZuJTS8M&w=419&h=206&ei =QmkuToahEsTRrQetpMWcAw&zoom=1&iact=hc&vpx=340&vpy=382&dur =451&hovh=80&hovw=163&tx=116&ty=113&page=4&tbnh=80&tbnw=163 &ved=1t:429,r:15,s:60&biw=1280&bih=551