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Game theory focuses on decision making involving participants, competitors, or other indiviudals in industries or markets for the best decision makeing, rather than making a decision in isolation. The Nash equilibrium helps economists understand how decisions that are good for the individual can be terrible for the group. Also, every person in a group makes the best decision for herself, based on what she thinks the others will do. And no-one can do better by changing strategy: every member of the group is doing as well as they possibly can. An example I came across in business is with two different companies competing. There is Company A and Company B and both companies want to determine whether they should launch a new advertising campaign for their products. If both companies start advertising, each company will attract 100 new customers. If only one company decides to advertise, it will attract 200 new customers, while the other company will not attract any new customers. If both companies decide not to advertise, none of the companies will engage new customers. Company A should advertise its products because the strategy provides a better payoff than the option of not advertising. The same situation exists for Company B. Thus, the scenario when both companies advertise their products is a Nash equilibrium.
Offer a reverse scenario?

ANSWER

Nash equilibrium is one of the fundamental concept of game theory, discovered by American mathematician John Nash.



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Question: prove by induction 2^2 + 4^2 + 6^2 + ... + (2n)^2 = (2n)(2n+1)(2n+2)/6 ANSWER we will use induction on n base case : n=1 we have, 2^2 = 2*3*4/6 = 4 which is true inductive hypothesis let it be true for n = k i.e.,  2^2 + 4^2 + ... + (2k)^2 =   [(2k)(2k+1)(2k+2)]/6 inductive case let n = k+1 then we have 2^2 + 4^2 + .... + (2k)^2 + (2(k+1))^2 =   [(2k)(2k+1)(2k+2)]/6 + (2k+2)^2 =(2k+2)*[(2k)(2k+1)/6 + (2k+2)] =(2k+2)*[ (4k^2+2k)/6 + (12k + 12)/6 ] =(2k+2)*[ (4k^2+14k+12)/6 ] = =(2k+2)*[(2k)(2k+1)/6 + (2k+2)] =(2k+2)*[ (4k^2+2k)/6 + (12k + 12)/6 ] =(2k+2)*[ (4k^2+14k+12)/6 ] = (2k+2)*[ (4k^2 + 8k + 6k + 12)/6 ] = (2k+2)*[ (4k(k + 2) +6(k+2))/6 ] = (2k+2)*[ (4k+6)(k+2)/6 ] =  (2k+2)*[ 2 (2k+3)(k+2)/6  ] =   (2k+2)*[  (2k+3)*2*(k+2)/6  ] =   (2k+2)*[  (2k+3)(2k+4)/6  ] = [(2*(k+1))(2*(k+1)+1)(2*(k+1)+2)]/6 replacing k+1 by m, we get replacing k+1 by m, we get [(2*m)(2*m+1)(2*m+2)]/6 this completes our proof b...

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